Patients who weigh < 10 kg are not eligible Patient with a BMI > 18 kg/m² and weighing at least 40 kg Patients must weigh ? 35 kg (due to possible size limitations with respect to percutaneous catheterization of the femoral artery and vein using the Delcath Hepatic Delivery System). Each unit must have a minimum of 1.5 x 10^7/kg pre-cryopreserved total nucleated cell dose. For non-red blood cell depleted units, the minimum pre-cryopreserved total nucleated cell dose of each unit must be at least 2.0 x10^7/kg. Cord blood unit(s) must be matched at a minimum of 4/6 to the recipient at HLA-A and B at low resolution using DNA-based typing and HLA-DRB1 at high resolution using DNA-based typing. Based on a published report from Eurocord, for single unit transplantation, the single unit pre-cryopreserved total nucleated cell (TNC) dose must be a minimum of 4.0 x10^7/kg recipient weight. For double cord transplants, each unit must have a minimum of 1.5 x10^7/kg pre-cryopreserved TNC and a minimum total of 4.0 x10^7/kg (sum of unit 1 and unit 2). For non-red blood cell depleted units, the minimum pre-cryopreserved TNC dose is 2.0 x10^7/kg recipient weight. or Subjects must be at least 12 kg or 24 pounds to be eligible for stem cell donation Subjects must be at least 12 kg or 24 pounds to be eligible for stem cell donation Patients must weigh > (25 kg) at the time of study entry Patients who weigh >= 50 kg At least 15 kg ELIGIBILITY FOR SCREENING: Weighs at least 12 kg Must be >= 10 kg Collection of an adequate number of CD34+ stem cells, i.e. >= 4-6 x 10^6/kg from apheresis Have at least 3 million x 10e6 CD34+ cells/kg to be infused Patients must weigh > 25 kg at the time of study entry Inability to purify >= 2.5 x 10^6 CD34-enriched cells/kg of patient weight from the pooled G-CSF mobilized leukapheresis products Has sufficient CD19 CAR T cell product (1x10^6 CAR T cells/kg) meeting release criteria for infusion Has sufficient CD19 CAR T cell product (1x10^6 CAR T cells/kg) meeting release criteria for re-infusion Weigh >=40 kg. > 40 kg Subjects who weigh less than 45 kg Patients must be >= 15 kg Less than or equal to 40 kg Minimum frozen PBSCs of 2 x 10^6 cluster of differentiation (CD)34 cells/kg for each transplant are mandatory and a PBSC of 2 x 10^6 CD34 cells/kg for back-up are strongly recommended (thus, PBSC of no less than 6 x 10^6 CD34 cells/kg is encouraged); these must all be collected prior to the initiation of consolidation Patients must weigh greater than 30 kg Must weigh greater than or equal to 20 kg. All patients eligible for therapeutic study must have (>= 2 x 10^6 CD34/kg) autologous hematopoietic stem cells harvested and cryopreserved Patients who receive greater than 12 mCi/kg are required to have stem cell rescue products harvested prior to study treatment; a minimum frozen autologous peripheral blood stem cell (PBSC) collection of 4 x 10^6 cluster of differentiation (CD)34+ cells/kg as 2 aliquots is the suggested dose; for subjects receiving < 12 mCi/ kg, a backup of 2.0 X 10^6 viable CD34+ cells/kg purged or unpurged PBSC is strongly recommended but not required Patients will be enrolled at collection of at least 3.0 x 10^6 CD34 cells/kg of autologous hematopoietic progenitor cells (HPC-A) by apheresis; a minimum of 2 collection procedures is required, unless collection on day # 1 > 5.0 x 10^6, CD34 cells/kg; a maximum of 10 collections is allowed; bone marrow harvest to supplement apheresis is not allowed Prior failed (< 5 x 10^6 CD34/kg) peripheral blood stem cell (PBSC) collection Patients weighing less than 50 Kg If a patient does not have a hematopoietic stem cell product available for re-infusion after MIBG treatment, they may not receive a 131I-MIBG dose > 12mCi/kg; patients must have a hematopoietic stem cell product available for reinfusion after MIBG treatment at doses of > 12 mCi/kg; the minimum quantity for peripheral blood stem cells is 1.5 x 10^6 cluster of differentiation [CD]34+ cells/kg (optimum > 2 x 10^6 CD34+ cells/kg); the minimum dose for bone marrow is 1.0 x 10^8 mononuclear cells/kg (optimum > 2.0 x 10^8 mononuclear cells/kg) Weight >40 kg (88 lb) and ?150 kg (330 lb); if between 13 and 14 years of age must weigh >= 50 kg (110 lb) Must have at least one additional aliquot of >= 1 x 10^6 CD34/kg cryopreserved cells stored at the time of transplant Patients must be >= 15 kg Must be 2 - 45 years of age and at least 10 kg Weighs at least 12kg Patient must weigh a minimum of 8 kg Patients will be enrolled after receiving at least two cycles of salvage cytoreductive chemotherapy and collection of at least 3.0 x 10^6 CD34 cells/kg of autologous hematopoietic progenitor cells (HPC-A) by apheresis; a minimum of 2 collection procedures is required, unless collection on day #1 > 5.0 x 10^6 CD34 cells/kg; a maximum of 10 collections is allowed; bone marrow harvest to supplement apheresis is not allowed Patients must weigh ? 35 kg (due to possible size limitations with respect to percutaneous catheterization of the femoral artery and vein using the Delcath Hepatic Delivery System). DONOR: Selection of two CB units is mandatory when a single cord blood unit does not meet the following criteria in the table below\r\n* Match grade \r\n** 6/6\r\n*** Single unit allowed for total nucleated cell (TNC) dose >= 2.5 x 10^7/kg\r\n** 5/6, 4/6\r\n*** Single unit allowed for TNC dose >= 4.0 (+/- 0.5) x 10^7/kg\r\n* If two CB units are used, the total cell dose of the combined units must be at least 3.0 x 10^7 TNC per kilogram recipient weight based on pre-cryopreservation numbers, with each CB unit containing a MINIMUM of 1.5 x 10^7 TNC/kg DONOR: The minimum recommended CD34/kg cell dose should be 2 x 10^5 CD34/kg, total dose from a single or combined double DONOR: Cord blood (CB) donor selection will be based on institutional guidelines and in general should be selected to optimize both human leukocyte antigen (HLA) match and cell dose; additionally, CB grafts shall consist of one or two CB donors based on, but not exclusively determined by, cell dose (total nucleated cell [TNC]/kg and CD34/kg), HLA matching and disease status and indication for transplant; attending preference will be allowed for single versus double unit as well as the degree of mismatching based on patient specific factors, as long as the following minimum criteria are met:\r\n* HLA matching\r\n** Minimum requirement: The CB graft(s) must be matched at a minimum at 4/6 HLA-A, B, DRB1 loci with the recipient. Therefore 0-2 mismatches at the A or B or DRB1 loci based on intermediate resolution A, B antigen and DRB1 allele typing for determination of HLA-match is allowed\r\n** HLA-matching determined by high-resolution typing is allowed per institutional guidelines as long as the minimum criteria are met\r\n* Selection of two CB units is mandatory when a single cord blood unit does not meet the following criteria:\r\n** Match grade 6/6; TNC Dose >= 2.5 x 10^7/kg\r\n** Match grade 5/6 or 4/6; TNC dose >= 4.0 (+/- 0.5) x 10^7/kg\r\n* If two CB units are used, the total cell dose of the combined units must be at least 3.0 x 10^7 TNC per kilogram recipient weight based on pre-cryopreservation numbers, with each CB unit containing a MINIMUM of 1.5 x 10^7 TNC/kg \r\n* The minimum recommended CD34/kg cell dose should be 2 x 10^5 CD34/kg, total dose from a single or combined double\r\n* The unmanipulated CB unit(s) will be Food and Drug Administration (FDA) licensed or will be obtained under a separate investigational new drug (IND), such as the National Marrow Donor Program (NMDP) Protocol 10-CBA conducted under BB IND-7555 or another IND sponsored by (1) a participating institution or (2) an investigator at FHCRC or one of the participating institutions\r\n* FHCRC only: Up to 5% of cord blood product, when ready for infusion, may be withheld for research purposes as long as thresholds for infused TNC dose are met; threshold for double unit transplantation is >= 3.0 x 10^7/kg; these products will be used to conduct studies involving the immunobiology of double cord transplantation and kinetics of engraftment Must have ? 3 x 10^8 NCs/kg BW OR 3 x 10^6/kg BW CD34-positive cells available for transplantation Total Collection of >= 4 x 10^6 CD34 cells/kg prior to transplant one INCLUSION CRITERIA FOR CCT: patients must have an autologous hematopoietic stem cell product cryopreserved and available for re-infusion after MIBG treatment; the minimum dose for peripheral blood stem cells is 2 x 10^6 CD34+ cells/kg Adequate cell dose > 2.5x10^6 CD34+ cells/kg The minimum dose for peripheral blood stem cells is: PURGED PBSC: 2.0 x 10^6 viable CD34+ cells/kg; UNPURGED PBSC: 2.0 x 10^6 CD34+ cells/kg (immunocytology is not required for peripheral blood stem cells) Availability of autologous peripheral blood stem cell graft, containing at least 6.0 x 10^6 CD34+ cells/kg DONOR: The CB unit selected for transplant must have a MINIMUM of 2.5 x 10^7 TNC/kg DONOR: The minimum recommended CD34/kg cell dose is 1.7 x 10^5 CD34/kg Prior apheresis of >= 3 million CD34+ cells/Kg. The subject must weigh >= 35 kilogram (kg). Participants who weigh > 110 kg will be ineligible due to study drug limitations Subjects must have collected at least 5 x 10^6 CD34+ cells/kg by apheresis after cycle 4 Has had a successful peripheral blood stem cell collection with G-CSF (filgrastim) +/- plerixafor (Mozobil) only; the target cell dose is >= 2.0 x10^6 CD34+ cells/kg All patients eligible for therapeutic study must have a minimum of >= 2 x 10^6 CD34/kg autologous hematopoietic stem cells harvested and cryopreserved At least 15 kg All patients eligible for therapeutic study must have a minimum of >= 4 x10^6 CD34/kg autologous hematopoietic stem cells harvested and cryopreserved and divided into 2 aliquots of at least >= 2 x10^6 CD34/kg each; patients with a history of prior autologous hematopoietic cell transplant (HCT) are only required to have >= 2x10^6 CD34/kg stored Patients must have a dose of unpurged peripheral blood stem cells is 2.0 x 106 viable CD34+ cells/kg available. Age of 12 years or older (patient must weigh ? 40 kg) Weigh >55 kg All patients eligible for therapeutic study must have a minimum of >= 2 x10^6 CD34/kg autologous hematopoietic stem cells harvested and cryopreserved Available autologous stem cells: >= 2 x 10^6 CD34+ cells/kg Patients weighing less than 30 kg Patient must be > 40 kg Patients who weigh >= 70 kg must be discussed with the principal investigator prior to enrolling on the protocol Subject must weigh at least 20 kg Patients will be treated at the same dose of tremelimumab as they previously received; for patients on dose level 1 who had already satisfied criteria for escalation to 10 mg/kg they will be re-treated at 10mg/kg Age 15 and above and >40 kg. Subjects must have a confirmed diagnosis of beta-thalassemia major and have been enrolled in a hypertransfusion program with a confirmed annual transfusion of >= 100 mL/kg/yr but < 200 mL/kg/yr, AND >= 8 transfusions of blood per year over a minimum of two years Diagnosis of CML except patients who have evidence of residual or persistent disease (morphologic, cytogenetic, or molecular) after a prior donor leukocyte infusion with a minimum cell dose of 1 x 10^8 cells/kg Prior allogeneic or autologous stem cell transplant using a myeloablative busulfan or total body radiation containing conditioning regimen defined as busulfan-based using a total dose of >= 12 mg/kg given by mouth or >= 10 mg/kg given IV; or a total-body irradiation (> 4 Gy) Subjects in whom the minimum stem cell dose of 5.0 x 10^6 CD34+ cells/kg has been collected PRIOR TO HIGH-DOSE CHEMOTHERAPY: Minimum of 2 x10^6 CD34+ cells/kg collected at mobilization Able to collect >= 1.5 x 10^6 CD34+/kg cell for transplantation The donor has a BW of at least 50 kg. No contraindication to the collection of a minimum of 4 x 10^6 CD34+ cells/kg by apheresis ELIGIBILITY FOR 2.4 MG/KG DOSING IN THE NEW COHORT > 10 kg at the time of apheresis; patients between 10-15 kg. must be approved by the apheresis unit prior to enrollment on protocol Skeletally mature adolescents must weigh at least 45 kg Patients weighing < 40 kg Patients must be undergoing a myeloablative allogeneic hematopoietic cell transplant with one of the following conditioning regimens:\r\n* Busulfan (>= 12.8 mg/kg IV or PO) and cyclophosphamide (>= 120 mg/kg)\r\n* Total body irradiation (TBI) (>= 1200 cGy) and etoposide (60 mg/kg)\r\n* TBI (>= 1200 cGy) and cyclophosphamide (120 mg/kg) Patients < 55 kg or > 140 kg, based on literature regarding accuracy of FloTrac Overweight or obese (> 25 kg/m^2) Patients whose BMI falls outside the eligible range (< 25 kg/m^2 or > 33 kg/m^2) Must weigh 20 kg Patients weighing < 50 kg The patient must weigh less than 150 kg (330 lb), which is the limit of the imaging couch Males and females must weigh >= 40 Kg Weigh more than 18 kg Patients who weigh > 70 kg are excluded; in addition, patients who weigh > 136 kg are excluded Patient weighs > 30 kg Subject has confirmed sufficient expression of NY-ESO-1 and/or LAGE-1a by reverse transcription polymerase chain reaction (RT-PCR) as determined by a central laboratory contracted by the Sponsor (this determination will also be made under a pre-enrollment screening ICF). Tumor expression of NY-ESO-1 or LAGE-1 by immunohistochemistry (IHC) and/or reverse transcriptase polymerase chain reaction (RTPCR) NY-ESO-1 positive by immunohistochemistry (IHC) utilizing commercially available NY-ESO-1 antibodies No prior treatment with a hypomethylating agent, or previous exposure to DEC-205/NY-ESO-1 fusion protein CDX-1401 (CDX-1401) NY-ESO-1 positive malignancy by immunohistochemistry (IHC) utilizing commercially available NY-ESO-1 antibodies NY-ESO-1 positive malignancy by immunohistochemistry (IHC) utilizing commonly available NY-ESO-1 antibodies Measurable metastatic cancer that expresses NY ESO-1 as assessed by one of the following methods: reverse transcriptase-polymerase chain reaction (RT-PCR) on tumor tissue, or by immunohistochemistry of resected tissue, or serum antibody reactive with ESO For cohorts 1, 2 and 3 only: Patient's tumor must be positive by histological or molecular assay for NY-ESO-1, according to the screening algorithm; historical results may be used\r\n* For cohort 4, NY-ESO-1 results will be noted but NY-ESO-1 positivity is not required for eligibility Measurable metastatic cancer or locally advanced refractory/recurrent malignancy including melanoma that expresses ESO as assessed by one of the following methods: reverse transcriptase-polymerase chain reaction (RT-PCR) on tumor tissue, or by immunohistochemistry of resected tissue, or serum antibody reactive with ESO NY-ESO-1 positive malignancy by immunohistochemistry (IHC) utilizing commonly available NY-ESO-1 antibodies Expression of one (1) or more of the following TAPAs: Sp17, AKAP-4, Ropporin, PTTG-1, Span-xb, Her-2/neu, HM1.24, NY-ESO-1 and MAGE-1, by either RT-PCR and/or immunocytochemistry, Western blotting or ELISA, in neoplastic cells and/or blood. For HER-2/neu expression, positive FISH results are acceptable. Immunohistochemistry (IHC) results from tumor biopsy for NY-ESO-1 positive Have received prior NY-ESO-1 therapy Tumor expression of NY-ESO-1 (2+ staining or > 25%) by immunohistochemistry (IHC). Prior administration of other NY-ESO-1 targeting immunotherapeutics. Tissue available from primary and/or recurrent disease to evaluate tumor expression of NY-ESO-1 or PDL1 by immunohistochemistry (IHC) and/or reverse transcriptase-polymerase chain reaction (RT-PCR), and for measurement of DNA methylation No requirement for tumor expression of NY-ESO-1 Patients with fully resected stage IIb through IV melanoma, with melanoma validated by histology or cytology, who have NOT received prior therapy\r\n* Patients may have had primary cutaneous, mucosal, or ocular melanoma or metastasis from an unknown primary site\r\n* Tissue should be submitted for evaluation of NY-ESO-1 expression and T-cell infiltrates; however, availability of tissue and/or positivity for NY-ESO-1 is not mandatory Positive staining of the most recently resected tumor tissue with antibodies to 1 or more of the following: human melanoma black (HMB) 45 for glycoprotein (gp) 100, NY-ESO-1, and/or melanoma-associated antigen recognized by T cells (MART)-1 Positive staining of tumor tissue with antibodies to 1 or more of the following: human melanoma black (HMB) 45 for gp100, or cancer-testis antigen (NY-ESO-1) Tumor specimen positive for NY-ESO-1 expression by IHC. Prior administration of other NY-ESO-1-targeting immunotherapeutics. Tumor specimen positive for NY-ESO-1 expression by IHC and/or RT-PCR. At least one tumor must be accessible and patients must consent for biopsies in Arms C and D. Prior administration of other NY-ESO-1-targeting immunotherapeutics Tumor histology consistent with melanoma tumor specimen positive for NY-ESO-1 expression by IHC and/or RT-PCR. Prior administration of other NY-ESO-1-targeting immunotherapeutics. NY-ESO-1 positive malignancy by immunohistochemistry (IHC) utilizing commonly available NY-ESO-1 antibodies Patients must have confirmed expression of NY-ESO-1 and/or LAGE-1 by RT-PCR, immunohistochemistry or quantigene analysis. (This determination will be made under a pre-enrollment screening ICF) Patient's tumor must be positive by histological assay for NY-ESO-1?²??T, according to the screening algorithm as described in Section 3.3.1. Positive expression is defined as ? 50% of cells that are 2+ and/or 3+ by immunohistochemistry For patients enrolled to Phase II Cohort B: Previous administration of vaccine therapy targeting NY-ESO-1. Measurable metastatic melanoma that expresses ESO as assessed by one of the following methods: reverse transcription polymerase chain reaction (RT-PCR) on tumor tissue, or by immunohistochemistry of resected tissue, or serum antibody reactive with ESO Tumor expression of NY-ESO-1 or LAGE-1 antigen by IHC or RT-PCR, or evidence of seropositivity to NY-ESO-1 or LAGE-1. Prior exposure to NY-ESO-1 vaccine. Positive NY-ESO-1 expression by reverse transcription (RT)-polymerase chain reaction (PCR) and/or immunohistochemistry (IHC) will be required for entry, as determined by analysis at the trial central laboratory Previous administration of vaccine therapy targeting NY-ESO-1 Patients must express NY-ESO-1 in their tumor by immunohistochemistry (IHC) (> 5%) prior to leukapheresis Tumor expression of NY-ESO-1 by immunohistochemistry (IHC) and/or real-time polymerase chain reaction (RTPCR) Patients may have received previous NY-ESO-1 vaccine therapy NY-ESO-1 expression in tumor by immunohistochemistry (IHC) is not required prior to screening consent; however, patients must have NY-ESO-1 expression to proceed with the leukapheresis; additionally patients must also meet the following criteria to proceed with leukapheresis (any exceptions to this will require prior approval by the Apheresis director and Principal Investigator): NY-ESO-1 positive by IHC (for this study, even a small level of positivity is acceptable); for patients with < 5% NY-ESO-1 by IHC positivity, the level of staining should be discussed with the patient and they should be informed that this will likely effect the efficacy of the therapy; this conversation must be documented in the patient's medical chart Patients must have NY-ESO-1 specific cells already produced or in production; these cells may be either in the process of their final expansion (for fresh infusion) or expanded and frozen at the time of enrollment Patients for whom we are unable to generate NY-ESO-1 specific cells Subject's tumor (either the most recent archival specimen or a fresh biopsy) shows positive NY-ESO-1 expression defined as ?30% of cells that are 2+ or 3+ by immunohistochemistry. All samples must have been pathologically reviewed by an Adaptimmune designated central laboratory. Patients must have proven positive tumor sample for NY-ESO-1 as follows: Cancer/testis antigen 1B (NY-ESO-1) specific T cell therapy within 1 year of starting on the trial Within 60 days after bone marrow biopsy confirmed remission after the patients recover from their last course of chemotherapy, the goal will be to consent the eligible patient prior to the remission confirmation bone marrow biopsy at the end of the planned chemotherapy); ideally, the research samples will be collected during the bone marrow biopsy, and the patient will be enrolled to the study within 2 weeks of the bone marrow biopsy; if there is delay to enroll the patient after the bone marrow biopsy and research sample collection, it is ok not to repeat bone marrow biopsy within 4 weeks, after the last bone marrow biopsy, if there is no sign of disease relapse; a repeat bone marrow biopsy should be done if the delay of enrollment is more than 4 weeks after the last bone marrow biopsy; patients with confirmed remission within 60 days after the last bone marrow biopsy, without research samples collection, should have a repeat bone marrow biopsy conducted within two weeks prior to enrolling on the study Bone marrow involvement (> 25%)\r\n* Note: bone marrow aspiration/biopsy can be performed at the time of diagnosis and does not need to be repeated unless there is a change in peripheral blood counts or it was performed more than 14 days prior to study entry Patients with bone marrow failure syndromes One of the following Ph-like ALL genetic lesions must be present in the diagnostic bone marrow or peripheral blood sample: STEP I: Patients must be diagnosed with symptomatic standard-risk multiple myeloma (SR-MM) as defined by all of the following (except gene expression profile [GEP]70 status if unknown):\r\n* No evidence of t(14;16) by fluorescence in situ hybridization (FISH) testing on bone marrow or not available\r\n* No evidence of t(14:20) by FISH testing on bone marrow or not available\r\n* No evidence of deletion 17p by FISH testing on bone marrow\r\n* FISH should be from within 90 days of registration\r\n** NOTE: If the FISH result states that no immunoglobulin heavy chain (IgH) abnormality is present, both t(14;16) and t(14;20) can be considered negative; in addition, if the patient has a t(11;14) or t(4;14) translocation present, they can be considered negative for t(14;16) and t(14;20); if testing for t(14;16) or t(14;20) could not be performed for lack of sufficient material or non-availability of the probe in the test panel, patients can be enrolled on the study\r\n* Standard Risk GEP70 signature within the past 90 days (only if GEP has been done and results are available)\r\n** NOTE: GEP testing is NOT a requirement for the study; if the test has been done, patients found to have a GEP70 status of high-risk will not be eligible\r\n* Serum lactate dehydrogenase (LDH) =< 2 x upper limit of normal (ULN) within the past 28 days\r\n* No more than 20% circulating plasma cells on peripheral blood smear differential or 2,000 plasma cells/microliter on white blood cell (WBC) differential of peripheral blood within the past 90 days\r\n** NOTE: This is NOT the plasma cell % from the marrow aspirate Diagnostic bone marrow and peripheral blood specimens must be submitted for immunophenotyping and selected molecular testing, and the establishment of BCR/ABL status; testing will be performed by the Eastern Cooperative Oncology Group (ECOG)-American College of Radiation Imaging Network (ACRIN) Leukemia Translational Research Laboratory (LTRL) and reported to the institution\r\n* NOTE: IT IS ESSENTIAL THAT A SAMPLE CONTAINING SUFFICIENT BLAST CELLS BE SUBMITTED TO THE ECOG-ACRIN LTRL AT BASELINE SO THAT SUBSEQUENT BONE MARROW ASSESSMENTS OF MRD CAN BE DONE; IN ADDITION TO ALLOWING THE LTRL TO CONFIRM ELIGIBILITY BASED ON BLAST CELL IMMUNOPHENOTYPE AND BCR/ABL STATUS, IT IS ALSO IMPERATIVE THAT AN ADEQUATE NUMBER OF BLASTS BE BANKED FOR ANALYSIS BY DRS MULLIGHAN/WILLMAN. WITHOUT ADEQUATE BASELINE SAMPLES, PATIENTS WILL NOT BE ABLE TO BE TREATED AND RANDOMIZED ON THIS PROTOCOL; IF A BONE MARROW ASPIRATE IS NOT AVAILABLE FOR LTRL SUBMISSION AT BASELINE, IT IS IMPERATIVE THAT DR PAIETTA FROM THE LTRL IS CALLED TO DISCUSS THE PERIPHERAL BLOOD WBC AND BLAST COUNT BEFORE BLOOD ONLY IS SUBMITTED\r\n* NOTE: Hydroxyurea can be given for up to 5 days prior to initiation of protocol therapy for control of leukocyte count and/or other symptoms or signs; corticosteroids can be given after pre-registration to the protocol and submission of baseline marrow and blood samples for control of leukocyte count and/or other symptoms or signs prior to initiation of protocol therapy if needed; if corticosteroids are given prior to pre-registration, contact the study chair as the patient may still be eligible to participate New diagnosis of B lineage ALL must be made upon bone marrow or peripheral blood immunophenotyping; cases with myeloid antigen expression, but unequivocal lymphoid immunophenotype, are eligible Mature B ALL (Burkitt’s-like leukemia) is excluded from enrollment in this trial; pre-study bone marrow biopsy and aspirate must be completed =< 1 week prior to registration Patients must have maintained peripheral blood evidence of a remission and must have a CR or CRi, confirmed on restaging bone marrow (BM) aspirate and biopsy Bone marrow aspirates must be submitted for centralized minimal residual disease (MRD) assessment performed by the ECOG-ACRIN Leukemia Translational Research Laboratory For patients pre-registering before the start of radiation therapy documentation of bone marrow aspirate and biopsy containing < 10% clonal plasma cells; radiation therapy should preferably begin within 28 days after bone marrow biopsy Bone marrow aspirate and biopsy containing < 10% clonal plasma cells performed after completion of RT and within 28 days prior to registration Patients must have bone marrow biopsy performed within 42 days prior to registration Patients must be newly diagnosed with a clinical diagnosis of APL (initially by morphology of bone marrow or peripheral blood)\r\n* Bone marrow is highly preferred but in cases where marrow cannot be obtained at diagnosis, peripheral blood will be accepted Patients with isolated myeloid sarcoma (myeloblastoma, chloroma, including leukemia cutis) but without evidence of APL by bone marrow or peripheral blood morphology are excluded All newly diagnosed patients must have evidence of ALL in their marrow or peripheral blood with at least 20% lymphoblasts present in blood or bone marrow collected within 28 days prior to registration; all relapsed/refractory patients (Cohort 2) must have at least 5% lymphoblasts present in blood or bone marrow collected within 28 days prior to registration; for relapsed/refractory patients, pathology and cytogenetics reports (both from time of original diagnosis) must be submitted at time of registration; if a bone marrow aspirate cannot be obtained despite an attempt (dry tap), appropriate immunohistochemistry (IHC) testing, including CD19, must be performed on the bone marrow biopsy to determine lineage; for ALL in marrow or peripheral blood, immunophenotyping of the blood or marrow lymphoblasts must be performed to determine lineage (B cell, T cell or mixed B/T cell); appropriate marker studies including cluster of differentiation (CD)19 (B cell), must be performed; co-expression of myeloid antigens (CD13 and CD33) will not exclude patients; if possible, the lineage specific markers (myeloid cells) should be determined; the blood/bone marrow sample for these assays must be obtained within 28 days prior to registration; patients with only extramedullary disease in the absence of bone marrow or blood involvement are not eligible Registration Step 3 – Maintenance: Patients must have documented CR or CRi within 28 days prior to registration; note that bone marrow examination is only required if there are clinical signs/symptoms of progression; if progression is a concern due to the length of the time for count recovery, a bone marrow examination is recommended Hemoglobin >= 8.0 g/dL for patients with solid tumors and known bone marrow metastatic disease Subjects without bone marrow metastases must have an ANC > 750/?l to begin treatment. Subject must be willing to provide fresh bone marrow samples during Screening (and prior to study treatment, if required). Bone marrow aspirate samples have been collected. Willing and able to undergo a pre-treatment bone marrow aspirate and biopsy and subsequent bone marrow aspirates and biopsies during treatment Specifically for participants in Arm A: Patient consent to collection of fresh bone marrow biopsy and aspirate for exploratory research obtained from a procedure performed no more than 28 days prior to initiating treatment on cycle 1, day 1 Not willing to undergo, not a candidate for, or not having a donor for immediate (within 3 months from the Screening date) bone marrow transplantation. Participants with any abnormal cytogenetics in the bone marrow not related to the lymphoma, and not deemed to be constitutional Patients are able and willing to provide bone marrow biopsies/aspirates as requested by the protocol Bone marrow specimen will be required at study entry; available deoxyribonucleic acid (DNA) sample from pre-induction bone marrow (BM) will be used for calibration step for MRD evaluation by gene sequencing A bone marrow biopsy must be performed within four weeks prior to cycle 1 day 1 treatment to establish the baseline fibrosis score, and consent is required prior to that bone marrow biopsy to assure tissue is collected for protocol mandated testing Bone marrow biopsy within 28 days of study drug infusion demonstrating at least 5% plasma cell involvement Receipt of radiotherapy to >25 % of bone marrow. Patients with relapsed or refractory SAA or very SAA defined:\r\n* Bone marrow (< 25% cellular)\r\n* Peripheral cytopenias (at least 2 of 3)\r\n** ANC < 500 per ml\r\n** Platelets < 20,000 per ml\r\n** Absolute reticulocytes (retic) < 60,000 or corrected retic < 1%\r\n* Very severe: as above, but ANC < 200\r\n* Disease may be designated as acquired or inherited if previous counts known (these other bone marrow failure disorders that are characterized by aplastic anemia may go by additional names such as dyskeratosis congenita or paroxysmal nocturnal hemoglobinuria [PNH])\r\n* Failed at least one course of immunosuppressive therapy (if presumed acquired disease); patients with inherited disease will be characterized as refractory and do not require immunosuppressive first If post allogeneic SCT must not have less than 50% donor chimerism in either peripheral blood or bone marrow Known bone marrow disorders which may interfere with bone marrow recovery or participants with delayed recovery from prior chemoradiotherapy ? 30% bone marrow blasts with at least 20% cellularity at mid-cycle bone marrow biopsy or residual AML on subsequent~ day 28 bone marrow biopsy by morphology, flow, PCR or FISH Agree to undergo a tumor/bone marrow biopsy of at least one metastatic site Suitable for imminent bone marrow transplant, or within 4 weeks of one. During the 4 weeks prior to inclusion in study, subjects must have a baseline bone marrow examination including all of the following:\r\n* Cytomorphology to confirm bone marrow blasts;\r\n* Cytogenetics; AND\r\n* Eastern Cooperative Oncology Group (ECOG) status 0-2 Bone marrow cellularity < 25% or marrow cellularity < 50% but with < 30% residual hematopoietic cells. In the haplo cohort, the potential donor must be willing to donate bone marrow. Have histologically or cytologically confirmed recurrent AML as defined by >= 5% myeloblasts by manual aspirate differential of bone marrow biopsy Able to provide bone marrow biopsy samples Bone marrow: > 25% donor T-cell chimerism in peripheral blood, obtained after 3 weeks post-transplant; ANC >1 x 10E9/L Myelodysplastic syndromes: diagnosis of very low or low risk MDS (biologically defined as low-risk MDS) by Revised-International Prognostic Score (R-IPSS), pathologically confirmed by a bone marrow aspirate and biopsy prior to registration; blast count must be < 20%\r\n* Bone marrow aspirate can be obtained from the subject at any time after the subject has given consent; the subject must be registered to the study within 30 days of obtaining the aspirate; (Note: if diagnostic bone marrow is obtained within 30 days prior to registration, a portion of the bone marrow aspirate collected may be used for research baseline sample to alleviate a second biopsy) Newly diagnosed lower GI grade II-IV aGVHD with clinical diagnosis based on modified Keystone criteria1 following allogeneic HSCT using bone marrow, peripheral blood stem cells, or cord blood. Grading of aGVHD will be based on International Bone Marrow Transplant Registry (IBMTR) criteria. < 25% bone marrow involvement of cellular marrow with lymphoma as determined by bilateral bone marrow aspirate and biopsy; NOTE: the percent involvement should be estimated by the hematopathologist using all of the biopsy material Marrow cellularity =< 15% (as determined on all bone marrow samples) Documentation of CT7, MAGE-A3, or WT1 expression in the bone marrow and/or bone marrow aspirate In morphologic remission (< 5% marrow blasts) based on bone marrow (BM) biopsy performed +/- 5 days of day 28 post-transplantation Bone marrow fibrosis that leads to a dry tap Known bone marrow dysplasia Evidence of relapsed/recurrent/residual disease as assessed by bone marrow aspirate and biopsy performed within 42 days prior to study entry Measurable MRD in bone marrow within 28 days prior to registration (MPF method) Leukocytes ? 750/mcL\r\n* If these cytopenias are not judged by the investigator to be due to underlying disease (i.e. potentially reversible with anti-neoplastic therapy); a subject will not be excluded because of pancytopenia ? grade 3 if it is due to disease, based on the results of bone marrow studies Platelets ? 50,000/mcL\r\n* If these cytopenias are not judged by the investigator to be due to underlying disease (i.e. potentially reversible with anti-neoplastic therapy); a subject will not be excluded because of pancytopenia ? grade 3 if it is due to disease, based on the results of bone marrow studies Evidence of myelodysplasia or cytogenetic abnormality indicative of myelodysplasia on any bone marrow biopsy prior to initiation of therapy. Diagnosis of myelofibrosis or other malignancy with moderate-severe bone marrow fibrosis. Myeloblasts account for less than 20% of leukocytes on peripheral blood and bone marrow aspirate Myeloblast count ? 20% in peripheral blood or bone marrow aspirate Hemoglobin ?8.0 g/dL (Grade ?2) maintained for ?1 week from any prior transfusion. Note: Grade ?3 neutropenia, thrombocytopenia, or anemia is permitted if the abnormality is related to bone marrow involvement with hematological malignancy (as documented by bone marrow biopsy/aspirate obtained since the last prior therapy). Adults up to 68 y/o with any of following: acute leukemia (ALL or AML), myelodysplasia, aplasia, and/or therapy (chemotherapy or radiation) induced bone marrow aplasia or hypoplasia with thrombocytopenia (platelet count ? 5,000 and ? 70,000/?L) for a minimum of 2 days. May include bone marrow transplant or peripheral or cord blood stem cell recipients, but not subjects with Graft-vs-Host disease. Patients with known bone marrow metastatic disease will not be eligible Patients must have > 20% plasma cells in the bone marrow aspirate differential <60 days prior to enrollment. The required bone marrow evaluation will need to be repeated for patients who received more than 1 cycle of anti-myeloma therapy (corticosteroid with or without other anti-myeloma agents) In subjects previously treated with blinatumomab, CD19 tumor expression in bone marrow or peripheral blood. Platelets >= 75,000/uL, if plasma cell percentage on bone marrow biopsy aspirate or core is > 30%, platelet eligibility requirement will be adjusted to 60,000/ul Patients with cytopenias attributable to bone marrow lymphomatous bone marrow involvement per current bone marrow biopsy may be enrolled if other inclusion criteria are met Bone marrow dysplasia Patients with newly diagnosed AML based on the World Health Organization classification who have persistent leukemia after a course or more of treatment with induction chemotherapy (the diagnosis of persistent disease, which is defined as > 10% blasts by evaluation of bone marrow biopsy or bone marrow aspirate) Be willing to provide tissue from bone marrow biopsies Be willing to provide tissue from a bone marrow biopsy if suspected involvement and/or lymph node at enrollment as well as a repeat bone marrow biopsy (if involved at diagnosis) after 3 of therapy and at the time of progression and/or completion of therapy whichever comes first Has acute promyelocytic leukemia as diagnosed by morphologic examination of bone marrow, by fluorescent in situ hybridization or cytogenetics of peripheral blood or bone marrow, or by other accepted analysis. Another bone marrow malignancy Bone marrow dysplasia Documented complete remission with full donor engraftment (by short tandem repeat [STR] identity testing) on day +30 bone marrow biopsy Confirmed histologic diagnosis on bone marrow biopsy and aspirate within 14 days of trial entry prior to starting cycle 1. Greater than 50% lymphoblasts on screening bone marrow aspirate or biopsy Corticosteroids and hydroxyurea are permitted after screening bone marrow biopsy is performed and for up to 7 days prior to starting study therapy During the 8 weeks prior to inclusion in study, subjects must have a baseline bone marrow examination including all of the following:\r\n* Cytomorphology to confirm bone marrow blasts\r\n* Cytogenetics > 10% bone marrow blasts at transplant if no history of AML and > 5% if had previous progression to AML (in a bone marrow biopsy 4 weeks prior to start of conditioning on study) All patients must be willing to undergo a mandatory serial bone marrow aspirate and/or biopsy at screening and subsequently following treatment for the assessment of biomarker/pharmacodynamics and disease status. Exceptions may be considered after documented discussion with Novartis. Bone marrow aspirate after completion of therapy demonstrates detectable DTCs (via IHC) Hemoglobin >= 8.0 g/dL – may be waived if abnormalities are due to disease related bone marrow involvement, within 14 days of study registration (within 30 days for pulmonary and cardiac assessments) Patients with prolonged pre-existing hematological toxicities including known indicators of bone marrow failure or abnormality CELL PROCUREMENT: Relapsed or refractory precursor B cell ALL:\r\n* Second or greater bone marrow relapse OR\r\n* Any bone marrow relapse >100 days after allogeneic stem cell transplant OR\r\n* Primary refractory ALL defined as no complete response after 2 cycles of a standard of care chemotherapy regimen OR\r\n* For adult subjects: first bone marrow relapse with duration of first complete response (CR) < 1 year OR CR1 duration >= 1 year and refractory to >= 1 cycle of therapy for treatment of relapse\r\n* Subjects with isolated non-central nervous system (CNS) extramedullary disease will be eligible as long as the time-of-remission criteria above for bone marrow relapses or primary refractory ALL are met and the biopsy for extramedullary disease confirms CD19 expression\r\n* For pediatric subjects: first bone marrow or isolated non-CNS extramedullary relapse refractory to 1 cycle of standard therapy for relapsed ALL\r\n* While active CNS3 leukemia will be excluded, subjects with concurrent CNS3 disease and bone marrow relapse who have responded to CNS-directed therapy prior to enrollment will be allowed to participate; intrathecal chemotherapy will be allowed to continue between lymphodepleting chemotherapy and cell infusion\r\n* Subjects with CNS2 disease and concurrent bone marrow relapse will be eligible; intrathecal chemotherapy will be allowed to continue between lymphodepleting chemotherapy and cell infusion Presence of >= 10% blast by morphologic examination of bone marrow aspirate or biopsy Outside diagnostic material is acceptable as long as peripheral blood and/or bone marrow slides are reviewed at the study institution; flow cytometric analysis of peripheral blood and/or bone marrow should be performed according to institutional practice guidelines Bone marrow aspirates/biopsies should be performed within 30 +/- 3 days from registration to confirm disease remission status Unwilling or unable to undergo serial bone marrow aspirate/biopsy Persisting (> 8 weeks) severe pancytopenia due to hematologic disorder or due to previous therapy rather than disease (ANC < 0.5 x 109/L or platelets < 30 x 109/L) - to be confirmed via bone marrow biopsy, as part of normal clinical care, prior to signing of consent. Patients with bone marrow metastatic disease will not be eligible Confirmed diagnosis of SAA (acquired or inherited), either from initial diagnosis or follow-up assessments, defined as:\r\n* Bone marrow hypocellularity is required and relative to patient’s age (normocellularity is 100- patient age in years)\r\n** Often marrow cellularity < 50% but with < 30% residual hematopoietic cells may be applied where appropriate at the discretion of the principal investigator (PI)\r\n* Two out of three of the following (in peripheral blood): neutrophils < 0.5 x 10^9/L, platelets < 20 x 10^9/L (without transfusions), reticulocyte count < 60 x 10^9/L The potential donor must be willing to donate bone marrow Isolated extramedullary leukemia without also meeting bone marrow criteria for acute leukemia (ie, ?20% blasts in bone marrow aspirate) Elevated expression above of WT1 in pre-treatment bone marrow or peripheral blood by either of two methods:\r\n* Increased expression of WT1 determined if the number of copies of WT1 divided by the number of copies of ABL x 10^4 is > 250 for bone marrow, or > 50 for peripheral blood; \r\n* Demonstration of WT1 overexpression will be determined by immunohistochemical staining (IHC) in blasts as compared to adjacent normal myeloid and erythroid precursors, as determined by a Fred Hutchinson/Seattle Cancer Care Alliance pathologist ELIGIBILITY FOR APHERESIS/BLOOD COLLECTION:\r\n* Human leukocyte antigen (HLA)-A*02:01 expression\r\n* Elevated expression above of WT1 in bone marrow or peripheral blood: increased expression of WT1 will be determined if the number of copies of WT1 divided by the number of copies of ABL x 10^4 is > 250 for bone marrow, or > 50 for peripheral blood; or when available, demonstration of WT1 overexpression will be determined by immunohistochemical staining (IHC) in blasts as compared to adjacent normal myeloid and erythroid precursors, as determined by Fred Hutchinson/Seattle Cancer Care Alliance hematopathology Elevated expression above baseline of WT1 in bone marrow or peripheral blood Either bone marrow or peripheral blood is allowed Outside diagnostic material is acceptable as long as peripheral blood and/or bone marrow slides are reviewed at the study institution by appropriate clinical staff; flow cytometric analysis of peripheral blood and/or bone marrow should be performed according to institutional practice guidelines High risk AML and MDS patients will be included; cohort 1: morphological relapse after stem cell transplant: \r\n* MDS patients: re-appearance of dysplastic changes in the bone marrow, with or without increase in bone marrow last count, which is pathologically consistent with myelodysplastic syndrome; \r\n* AML patients: bone marrow blast count >= 5% Congenital bone marrow failure syndrome Bone marrow tumour infiltration <25% tumour cells. Subjects must be able and willing to provide bone marrow biopsies/aspirates as requested by the protocol Patients with disease only in the bone may not have received Xofigo/radium 223 to avoid ongoing DNA damage in bone marrow Depressed bone marrow ANC ? 750 - cannot be transfused (must be ? 72 hours from last neutrophil infusion) However, the plt and ANC requirements can be waived if low counts thought to be secondary to leukemia or tumor bone marrow infiltration Subjects must have bone marrow with >= 5% blasts (M2 or M3 marrow) definitively identified either on a bone marrow aspirate or biopsy sample, as assessed by morphology, immunohistochemical studies, flow cytometry, karyotype, cytogenetic testing such as fluorescent in situ hybridization (FISH) or other molecular studies Bone marrow depression or hematologic parameters in the range that would increase the risk for severe bleeding Gastrointestinal or bone marrow or spleen only patients are allowable Non-secretory disease measurable with bone marrow biopsy or radiography. Myelodysplastic syndrome. If clinically significant anemia or pancytopenia exists, documentation of a bone marrow aspirate within 24 months prior to the first dose of PRTX-100 showing no evidence of myelodysplasia is required. Patients with solid tumors not metastatic to bone marrow: Bone marrow specimen from diagnosis (or pre-induction) will be required at study entry; available deoxyribonucleic acid (DNA) sample will be used for calibration step for MRD evaluation by gene sequencing Multiple myeloma in complete remission is defined as per Durie BG et al.:\r\n* Negative immunofixation on the serum and urine and disappearance of any soft tissue plasmacytomas and =< 5% plasma cells in the bone marrow; CR requires two consecutive assessments by serum and urine immunofixation made at any time prior to enrollment; CR also requires no known evidence of progressive or new bone lesions if radiographic studies are performed; confirmation with repeat bone marrow is not needed Received previous radiotherapy to approximately > 25% of bone marrow Bone marrow (BM) harvest required to reach adequate cell dose for transplant INCLUSION CRITERIA FOR ADDITIONAL PTCy-MILs INFUSION AT RELAPSE: Donor CD3+ chimerism >= 30% measured in peripheral blood or bone marrow No option for immediate bone marrow transplant unless patient refuses this therapy To be eligible for this protocol the patient must have AML not in remission defined as greater than >= 5% myeloblasts by aspirate morphology as determined by a bone marrow aspirate and biopsy obtained within 2 weeks of study registration\r\n* In the event induction treatment results in a hypoplastic bone marrow status (< 10% cellularity), precluding accurate enumeration of blast percentages, the patient is still eligible if the preceding bone marrow aspirate contained >= 5% myeloblasts; to meet this condition, prior induction therapy must have been completed a minimum of 21 days prior to this result Able and willing to provide bone marrow biopsies/aspirates as requested by the protocol. Bone marrow with tumor cells seen on routine morphology Patients with a diagnosis of acute myeloid leukemia (AML), acute lymphoblastic leukemia (ALL), or myelodysplastic syndromes (MDS) with fewer than 10% myeloblasts or lymphoblasts in the bone marrow and no blasts in the peripheral blood on morphologic analysis performed within 30 days of start of the conditioning regimen; if remission bone marrow is available beyond 30 days a new bone marrow evaluation is required to assess remission status\r\n* The diagnosis of AML, ALL, or MDS will be based on World Health Organization (WHO) criteria\r\n* Pre-transplant bone marrow sample must be evaluable for assessment of remission status (i.e. aspirate smear containing particles and/or evaluable bone marrow core biopsy)\r\n* Patients with leukemia infiltration in the central nervous system (CNS) are eligible if cerebrospinal fluid (CSF) cytospin is negative for myeloblasts or lymphoblasts at time of enrollment\r\n* If the patient has an intra-abdominal chloroma on presentation, and has a partial response or complete response to treatment (size reduction of chloroma and marrow blast < 10%), the patient is eligible; however the chloroma must be included as part of the treatment target Relapsed or refractory AML as defined by one of the following criteria:\r\n* First relapse within 12 months after date of first complete response (CR) or complete response with incomplete bone marrow recovery (CRi)\r\n* Persistent AML documented by bone marrow biopsy at least 28 days after day 1 of the first induction cycle of cytotoxic chemotherapy\r\n* Hypercellular bone marrow with greater than 20% cellularity and 10% blasts at least 14 days after first induction cycle day 1 Bone marrow with >= 5% lymphoblasts Histologic proof of neuroblastoma or positive bone marrow for tumor cells with increased urine catecholamines Patients must have documented WT1 positive disease; for purpose of this study, this is defined as detectable presence of WT1 expression by immunohistochemistry or by WT1 transcript via real time-polymerase chain reaction (RT-PCR) on a bone marrow or other plasma cell-related biopsy specimen prior to autologous stem cell transplantation; bone marrow or other biopsy specimen from time of diagnosis from patients diagnosed at MSKCC or outside hospital may be requested for assessment of WT1 expression by IHC Diagnosis of AML according to World Health Organization (WHO) diagnostic criteria (at least 20% blasts in the peripheral blood or bone marrow), with French-American-British Cooperative group (FAB) classification other than M3 (acute promyelocytic leukemia), documented by bone marrow aspiration and biopsy performed within 14 days prior to administration of 1st dose of remission induction chemotherapy; if a bone marrow aspirate and biopsy were obtained within 28 days prior to the first dose of remission induction therapy then these tests may be submitted for review at University of Southern California (USC) and a repeat screening bone marrow does not need to be conducted No morphologic evidence of leukemia or active MDS as determined by JHH hematopathologist independent review of a bone marrow aspirate and biopsy done following the completion of therapy Bone marrow hypocellular for age Patients must be able to provide blood and bone marrow samples and undergo the procedures required for this protocol In patients whose leukemic cells are available for evaluation, the expression of WT1 in the patient's bone marrow will be determined; if WT1 expression in the patient’s bone marrow is not highly expressed by polymerase chain reaction (PCR), the patient will be excluded from the study; patients with no evaluable leukemia will be eligible for enrollment based on the high frequency of positive leukemias (> 90%), and leukemia will be evaluated for WT1 expression if recurrence is detected Donor myeloid engraftment (from peripheral blood or bone marrow) of at least 40% documented =< 60 days from protocol therapy; a bone marrow engraftment analysis should show the cluster of differentiation (CD)15+ fraction to be at least 40% for inclusion No evidence of extranodal disease outside the chest including spleen and bone marrow. Bone marrow (BM) harvest required to reach adequate cell dose for transplant Willingness to undergo a pretreatment bone marrow biopsy and/or aspirate, or archival sample obtained since completion of most recent therapy (as appropriate to subjects with existing bone marrow disease or for whom bone marrow examination is a component of disease status assessment) Demonstrate NOXA BH3 priming of ?40% by mitochondrial profiling in bone marrow or 30 - 39% for NOXA Exploratory Arm. Evidence of poor bone marrow function (bone marrow cellularity less than 50% with at least one cytopenia) OR Recipients will have satisfactory organ function (excluding bone marrow) and will have a Karnofsky activity assessment > 90% and will have: Trephine biopsy is recommended (unless diagnosis can be confirmed by peripheral blood examination) in the event that bone marrow aspiration results in a \dry tap\ Diagnosis of hairy cell leukemia (HCL) established by bone marrow examination If recent mold infection e.g. Aspergillus - must have minimum of 30 days of appropriate treatment before bone marrow transplant (BMT) and infection controlled and be cleared by Infectious Disease Willing to provide research bone marrow aspirate specimen Both granulocyte-colony stimulating factor (G-CSF) stimulated peripheral blood grafts and bone marrow grafts will be the priority, although bone marrow graft source will be allowed based upon donor preference Presence of mutated BTK in ? 4% of peripheral blood or bone marrow CLL cells, or ?1% and rising on two separate measurements obtained at least 28 days apart. Primary bone marrow failure; Has a bone marrow examination performed within 14 days before baseline (C1D1). Prior radiotherapy in the adjuvant setting allowed provided that less than 25% of the bone marrow had been irradiated Patient with documented mastocytosis and evaluable disease based upon histological criteria: typical infiltrates of mast cells in a multifocal or diffuse pattern in skin and/or bone marrow biopsy Relapsed following autologous bone marrow transplantation (BMT), or are ineligible, or refused BMT Palliative bone-directed radiotherapy is permitted unless involving an area of ? 25% of bone marrow reserves and occurring within 5 weeks prior to the start of study treatment; 4. Sufficient and viable bone marrow aspirate or peripheral blood collection to use for the ex vivo sensitivity assay. Is able and willing to provide protocol-defined bone marrow biopsies/aspirates Inclusion Criteria for Cohort 2 in Part 2 only: Patients with >= 25% of the bone marrow radiated for other diseases Other bone marrow failure syndromes or low grade (< 5% bone marrow blasts) MDS Bone marrow biopsy (BMBx) within 28 days prior to first study treatment. Willing and able to undergo a pre-treatment bone marrow aspirate and biopsy and subsequent bone marrow aspirates and biopsies during treatment Newly-diagnosed AML patients according to World Health Organization (WHO) classification who are considered candidates for intensive chemotherapy based upon examination of peripheral blood or bone marrow aspirate specimens or touch preparations of the bone marrow biopsy obtained within two weeks prior to randomization; a bone marrow aspirate is required for enrollment; however, on occasion there is discordance between percentage of myeloblasts on the differential of the peripheral blood or aspirate; the peripheral blood criteria are sufficient for diagnosis; confirmatory immunophenotyping will be performed centrally Diagnostic bone marrow and peripheral blood specimens must be submitted for immunophenotyping and selected molecular testing Consolidation cycle 1 must commence within sixty days of the bone marrow aspirate and biopsy that confirmed the presence of a CR or CRi Patients must have maintained peripheral blood evidence of a remission and must have a CR or CRi, confirmed on restaging bone marrow (BM) aspirate and biopsy and cytogenetic analysis Subjects must be able and willing to provide bone marrow biopsies/aspirates and buccal mucosal sample as requested by the protocol Willing and able to undergo a pre-treatment bone marrow biopsy and subsequent on-treatment bone marrow biopsies Definitive radiotherapy (> 10 fractions and maximal area of hematopoietic active Bone Marrow treated greater than 25%) within 4 weeks prior to Screening Acquired bone marrow failure syndromes Congenital bone marrow failure syndrome DONOR: Able and willing to have up to 3 separate mobilized apheresis collections performed or if unable to undergo apheresis agrees to a bone marrow harvest (requires additional consent) Bone marrow plasma cells must make up 30% or less of total bone marrow cells based on a bone marrow biopsy performed within 30 days of the start of protocol treatment Patients who have not passed the nadir of bone marrow suppression from previous anti-myeloma therapy yet; if in doubt, serial complete blood counts (CBCs) with differential should be obtained Bone marrow aspirates/biopsies should be performed within 28 (+ 4 day window) days from registration to confirm disease remission status Bone marrow lymphoplasmacytosis with: Beta-human chorionic gonadotropin (B-HCG) will be performed on all women of child-bearing potential as screening prior to enrollment on this trial; signal transducer and activator of transcription 3 (STAT3) levels in the bone marrow will also be measured prior to enrollment; the B-HCG, STAT3 testing and bone marrow biopsy costs are not considered standard of care and have been included in the proposed budget A subject will not be excluded because of pancytopenia >= grade 3 if it is due to disease, based on the results of bone marrow studies Patients who received radiotherapy to more than 25% of their bone marrow Patients who received radiotherapy to more than 25% of their bone marrow Patients must have documented WT1 + disease; for purpose of this study, this is defined as detectable presence of any WT1 transcript via RT-PCR on a bone marrow performed at MSKCC within 4 weeks prior to the administration of the first dose of vaccine Pregnancy at the time of bone marrow transplant (BMT) Bone marrow dysplasia Patients must have a unilateral or bilateral bone marrow biopsy performed within 42 days prior to registration Diagnosed with PMF, PPV-MF or PET-MF as confirmed by bone marrow biopsy 2nd or greater Bone Marrow (BM) relapse OR Bone marrow with ? 5% lymphoblasts by morphologic assessment at screening 2nd or greater Bone Marrow (BM) relapse OR. Bone marrow with ? 5% lymphoblasts by morphologic assessment at screening. Must be willing to consent to two or more bone marrow aspirate procedures to be completed during study. Inaspirable bone marrow. quantifiable bone marrow infiltration documented in a bone marrow biopsy during screening Willing and able to undergo bone marrow aspirate per protocol (with or without bone marrow biopsy per institutional guidelines). At least Grade 2 marrow fibrosis according to the WHO Grading of Bone Marrow Fibrosis; Morphological or cytological features of myelodysplasia and/or post chemotherapy aplasia on bone marrow (BM) assessment. Myelodysplastic Syndrome (MDS) according to World Health Organization (WHO) criteria confirmed by bone marrow aspirate and biopsy within 12 weeks prior to first dose. A local laboratory report from this diagnostic bone marrow aspirate and biopsy must be approved by the sponsor RAEB 1 - defined as having 5% to 9% myeloblasts in the bone marrow. WHO defined AML with 20% to 30% myeloblasts in the bone marrow and <30% myeloblasts in peripheral blood who are deemed by the investigator to be appropriate for azacitidine based therapy. Ability to undergo the study required bone marrow sample collection procedures. Acute promyelocytic leukemia as diagnosed by morphologic examination of bone marrow, by fluorescent in situ hybridization or cytogenetics of peripheral blood or bone marrow, or by other accepted analysis. Outside diagnostic material is acceptable as long as peripheral blood and/or bone marrow slides are reviewed at the study institution; flow cytometric analysis of peripheral blood and/or bone marrow should be performed according to institutional practice guidelines Patients with elevated catecholamines (i.e., > 2 x ULN) only or bone marrow disease only are NOT eligible for this study Hematological values within the limits independent of growth factor support or transfusion unless cytopenia is caused by the underlying disease, i.e., no evidence of additional bone marrow dysfunction (e.g., myelodysplastic syndrome, hypoplastic bone marrow) Current use of any chemotherapy agent likely to cause myeloablation (severe or complete depletion of bone marrow). Subjects with clonal evolution in Ph+ cells observed in ?2 metaphases at baseline bone marrow cytogenetic test, unless the same abnormalities were present at diagnosis. Patients with no evidence of clonal evolution, including those patients whose cytogenetic testing fails or bone marrow aspiration is a dry tap at 3 months, are eligible for the study Bone marrow cellularity of >= 50% of age defined normal values by core biopsy; cellularity must be evaluated within 90 days of the dosimetry infusion and at least 21 days after receiving any cytoreductive/myelosuppressive chemotherapy Radiotherapy to ? 25% of the bone marrow within 4 weeks prior to randomization Product planned for infusion is bone marrow Subject consents to serial bone marrow aspiration and biopsies as specified. Must have measurable disease, including one of the following: absolute lymphocyte count greater than 5000/uL, lymphadenopathy greater than 1.5 cm in longest dimension, splenomegaly (palpable at least 1 cm below the costal margin or radiographically enlarged), bone marrow biopsy with residual CLL cells, or resultant bone marrow dysfunction (platelet count < 100 k/uL, hemoglobin < 10 g/dL) Patients with known metastatic tumor in the bone marrow Gastrointestinal or bone marrow or spleen only patients are allowable and will be analyzed separately Have a diagnosis of CLL based on peripheral blood flow cytometry and/or bone marrow aspiration and biopsy OR diagnosis of SLL based on lymph node or bone marrow biopsy; patients with SLL need to have measurable disease Platelets >= 75,000/uL, if plasma cell percentage on bone marrow biopsy aspirate or core is > 30%, platelet eligibility requirement will be adjusted to 60,000/uL For purposes of this study, bone marrow aspirate and biopsy are not considered surgical procedures and therefore are permitted within 14 days prior to start of protocol therapy Patients with Ewing sarcoma metastatic to the bone marrow are not required to meet bone marrow criteria for study eligibility and are not evaluable for hematologic toxicity For subjects in the Phase 2 portion of the trial, central testing of IDH2 mutation of bone marrow aspirate and peripheral blood, is required during screening to confirm eligibility The diagnosis and evaluation of AML or MDS will be made by bone marrow aspiration and/or biopsy. If an aspirate is unobtainable (i.e., a \dry tap\), the diagnosis may be made from the core biopsy. Screening bone marrow aspirate and peripheral blood samples are required of all subjects. A bone marrow biopsy must be collected if adequate aspirate is not attainable unless: A bone marrow aspirate and biopsy was performed as part of the standard of care within 28 days prior to the start of the study treatment; and Slides of bone marrow aspirate, biopsy and stained peripheral blood smear are available for both local and central pathology reviewers; and A bone marrow aspirate sample acquired within 28 days prior to the start of study treatment has been sent for cytogenetic analysis. Patients with relapsed/refractory disease must have morphologic proof (from bone marrow aspirate, smears or touch preps of bone marrow biopsy) of AML with >= 10% blasts within two weeks (14 days) prior to initiation of therapy\r\n* All immunophenotype and cytogenetic/molecular groups are eligible for participation except for acute promyelocytic leukemia (APL) (as proven by the presence of promyelocytic leukemia/retinoic acid receptor alpha [PML-retinoic acid-receptor-[RAR] alpha]) REGISTRATION INCLUSION CRITERIA: Presence of bone marrow ERBB2 overexpressing DTCs at the time of diagnosis; bone marrow aspiration will be performed in consented patients to evaluate DTCs following pre-registration provided patients meet all eligibility criteria AML with a history of CMMoL: CMMoLAML must have bone marrow documentation of prior CMMoL Patients must be available for follow-up evaluations at 30, 60, 180 days post bone marrow transplant (BMT) and yearly thereafter indefinitely Bone marrow myelodysplasia and/or chromosomal abnormalities Acute myeloid leukemia with a fms-like tyrosine kinase 3 (FLT3)-internal tandem duplication (ITD) who are in a complete remission or partial remission (less than 10% blasts in marrow) as documented by bone marrow biopsy and who plan to undergo a bone marrow transplantation Patients with lack of engraftment (less than 90% donor deoxyribonucleic acid [DNA] in bone marrow or peripheral blood) after bone marrow transplant as evidence by RFLP (restriction fragment length polymorphism) are not eligible Bone marrow with less than 15% lymphoma cells following salvage therapy; no evidence of myelodysplasia Evidence of myelodysplasia on any bone marrow biopsy ALL in first bone marrow relapse occurring > 18 months (> 540 days) from initial diagnosis; marrow must have >= 25% blasts (M3 marrow), either on an aspirate or biopsy sample, as assessed by morphology, flow cytometry, and/or immunohistochemistry; individuals with CNS, testicular or other extramedullary involvement are eligible as long as they also meet marrow involvement criteria A bone marrow aspiration performed within 21 days prior to the start of pre-infusion preparative therapy confirms the patient is in CR1 Presence of circulating malignant lymphoid cells or bone marrow with lymphoma that constituted more than 25% of the cellular elements Patients must have a bone marrow biopsy (bilateral preferred, unilateral acceptable) within 90 days prior to starting treatment Less than 5% marrow involvement with NHL within 4 weeks of study as defined by unilateral bone marrow aspiration and biopsy no more than 10% of the subject's bone marrow is irradiated Patients with >= 20% bone marrow involvement or plasmacytoma amenable to resection under local anesthesia Other bone marrow failure syndromes Testing for extraneural metastasis by bone scan or bone marrow biopsy will not be performed routinely on this protocol; in the unlikely event that extraneural metastasis is detected on an evaluation performed at an outside institution prior to referral or because of clinical suspicion, such M4 patients will be eligible for protocol treatment on the high-risk arm Recent bone marrow biopsy and cytogenetic analysis Detectable tumor on radiographic studies or bone marrow biopsy prior to mobilization regimen Bone marrow consistent with plasma cell dyscrasia bone marrow exam is performed at screening and demonstrates quantifiable CLL. Cytotoxic regimens are any that include agents that target the genetic and/or mitotic apparatus of the dividing cells, resulting in dose limiting toxicity to the bone marrow or gastrointestinal mucosa First or greater bone marrow relapse from CR, or Has already had a bone marrow biopsy and aspirate to assess remission status after induction therapy DONOR: Meets institutional selection criteria for organ and bone marrow donation Bone marrow biopsy must be negative for lymphoma. Patients should be eligible for transplantation according to the Bone Marrow Transplant (BMT) Policy Manual Requiring salvage chemotherapy for persistent/refractory or relapsed disease after at least one course of conventional chemotherapy, e.g. with “7+3”, as defined by persistence of >= 20% myeloid blasts on bone marrow aspirate or peripheral blood smear; a bone marrow biopsy is not routinely required, but should be obtained if the aspirate is dilute, hypocellular, or inaspirable; outside bone marrow examinations performed within the stipulated time period are acceptable for screening as long as the slides are reviewed at the study institution; flow cytometric analysis of the bone marrow aspirate should be performed according to institutional practice guidelines Palliative radiation therapy (RT) for metastatic disease is allowed only if =< 25% of total body bone marrow was irradiated and < or = 35Gy administered to the pericardial area; 28 days must have elapsed since completion of RT with bone marrow recovery; soft tissue disease irradiated in the prior 2 months may not be designated as measurable disease Monoclonal bone marrow plasmacytosis ?30% (evaluable disease) Measurable disease by one of the following: radiographic criteria (>= 2 cm by computed tomography); lymphoma involving peripheral blood with more than 5000 leukemia cells/mm^3, or any degree of bone marrow infiltration on bone marrow biopsy; skin involvement with or without nodal or bone marrow involvement permitted for cutaneous lymphomas is also permitted Prior pelvic radiation (e.g. external beam, brachytherapy, etc) that, in the opinion of the investigator, may lead to decreased bone marrow cellularity in a marrow sample obtained from a pelvic bone marrow biopsy Extensive radiotherapy (to greater than 15% of bone marrow) Ongoing anticoagulant therapy that cannot be held if necessary to permit bone marrow sampling. Participants with known bone marrow disorders which may interfere with bone marrow recovery or participants with delayed recovery from prior chemoradiotherapy Radiotherapy involving <25% of the hematopoietically active bone marrow within 21 days preceding first dose of study treatment Radiotherapy involving ?25% of the hematopoietically active bone marrow within 42 days preceding first dose of study treatment Prior radiotherapy to > 25% of bone marrow volume. Pathological confirmation by bone marrow documenting the following: All patients must be willing to undergo a mandatory bone marrow aspirate and/or biopsy at baseline for the assessment of biomarker/pharmacodynamics and disease status DONOR: Meets institutional selection criteria for organ and bone marrow donation Presence of phosphorylated p65 NF-kB component in at least 5% of bone marrow cells Patient must be able/willing to undergo bone marrow aspirate and biopsy Prior radiotherapy to >= 40% of bone marrow Able to undergo bone marrow aspiration and biopsy at screening Acute promyelocytic leukemia as diagnosed by morphologic examination of bone marrow, by fluorescent in situ hybridization or cytogenetics of peripheral blood or bone marrow, or by other accepted analysis Patients must have peripheral blood and bone marrow aspirate specimens obtained within 28 days prior to registration submitted for translational medicine; with patient consent, residuals will be banked for future research Relapsed/persistent disease according to standard criteria requiring salvage therapy; outside diagnostic material is acceptable as long as peripheral blood and/or bone marrow slides are reviewed at the study institution; flow cytometric analysis of peripheral blood and/or bone marrow should be performed according to institutional practice guidelines Patients with severe pancytopenia not meeting the above criteria for cell counts due to documented, extensive MM involvement of the bone marrow (suggested by >= 50% bone marrow plasmacytosis) will be eligible for enrollment Patients must have unilateral or bilateral bone marrow biopsy performed within 42 days prior to registration Any amount of neuroblastoma tumor cells in the bone marrow based on routine morphology (with or without immunocytochemistry) in at least one sample from bilateral aspirates and biopsies. For patients enrolling in Stage 2, the bone marrow evaluation determined locally within the previous 6 months indicates the presence of MRD. Unable to tolerate bone marrow biopsy Relapsed disease is defined as the reappearance of leukemia cells in the bone marrow or peripheral blood or elsewhere in the body (other tissues/organs) after the attainment of a CR. Available donor able to undergo a bone marrow harvest; for matched unrelated donor transplants only: peripheral blood stem cells may be collected if donor is unavailable for bone marrow harvest or if adequate bone marrow cannot be collected Substantial radiotherapy to the bone marrow within 6 weeks prior to enrollment. Bone marrow cellularity less than 25% or marrow cellularity less than 50% but with less than 30% residual hematopoietic cells At pre-ASCT evaluation patients must meet the International Neuroblastoma Response Criteria (INRC) for CR, VGPR, or PR for primary site, soft tissue metastases and bone metastases; patients who meet those criteria must also meet the protocol specified criteria for bone marrow response as outlined below:\r\n* =< 10% tumor (of total nucleated cellular content) seen on any specimen from a bilateral bone marrow aspirate/biopsy\r\n* Patient who have no tumor seen on the prior bone marrow, and then have =< 10% tumor on any of the bilateral marrow aspirate/biopsy specimens done at pre-ASCT and/or pre-enrollment evaluation will also be eligible (note that per INRC this would have been defined as “overall” response of progressive disease [PD]) Pathological confirmation by bone marrow documenting the following: Patients must have histologically confirmed diagnosis of Philadelphia-negative ALL by bone marrow biopsy or aspirate Patients must have >= 5% leukemic blasts in the bone marrow and/or increasing levels of MRD in the bone marrow as assessed by flow cytometry; if an adequate bone marrow sample cannot be obtained, patients may be enrolled if there is unequivocal evidence of leukemia in the peripheral blood Stage II Arm B: prior CDK4/6 inhibitor treatment, bone marrow transplant or extensive radiotherapy to ?25% of bone marrow Subjects must have presence of < 20% bone marrow blasts per bone marrow biopsy/aspirate at screening. No morphological evidence of disease (defined as marrow myeloblast percentage of < 5% and/or documentation from hematopathologist indicating no morphological evidence of leukemia) on day 14 bone marrow examination (range, day 14-17; day 1 refers to the first day of IC) following remission IC Any evidence of fibrosis on morphological examination of bone marrow at the time of AML diagnosis No evidence of recurrent disease on bone marrow evaluation done within 21 days of enrollment Presence of clinically significant bone marrow fibrosis on the bone marrow examination immediately prior to UCBT Patients undergoing additional procedures during the same anesthetic such as bone marrow aspirate or biopsy will be excluded Subjects who are scheduled for bone marrow ablation chemotherapy Patient has had prior bone marrow procedures Patient is receiving additional potentially painful interventions (e.g. central line insertion/removal) concurrent with the bone marrow procedure Patients with at least ONE of the following sites of measurable disease according to International Workshop Criteria87: A) Measurable tumor on MRI or CT scan. Measurable is defined as at least one lesion 20 mm in at least one dimension; for spiral CT, measurable is defined as 10 mm in at least one dimension. For patients with persistent disease, a biopsy of bone marrow, or bone, or a soft tissue site, must have demonstrated viable tumor. If lesion was radiated, biopsy must have been done at least 4 weeks after radiation completed. B) Bone marrow with tumor cells seen on routine morphology (not by NSE staining only) of bilateral aspirate and/or biopsy on one bone marrow sample, except for patient who tested positive subsequent to their last treatment regimen or patients who had a negative marrow within three months of study entry. Patients with known bone marrow reticulin fibrosis (>= grade 2) (only applicable to patients with CML) AML patients with persistent disease from the recent treatment defined as > 5% blast and/or > 20% cellularity and reported as persistent residual disease by a pathological report of the patient's bone marrow biopsy 14 days +/- 2 days from the initiation of cytarabine Bone marrow aspirates/biopsies should be performed within 23 ± 7 days from registration to confirm disease remission status Phase 1 only: known bone marrow fibrosis; Phase 2 only: Bone marrow fibrosis grade 2 or greater; Bilateral bone marrow aspirates and biopsy Able to begin study treatment between day 60 and day 100 after the transplant and meets the following transplant related requirements:\r\n* Sustained neutrophil (absolute neutrophil count [ANC] > 1000/mcL) and platelet (> 30,000/mcL) engraftment\r\n* > 50% donor chimerism in blood or bone marrow\r\n* No evidence of recurrent disease on most recent bone marrow evaluation (day 21 or 28 post-transplant is acceptable)\r\n* No morphologic evidence of relapse (< 5% bone marrow blasts)\r\n* Ability to be treated in the outpatient setting (not an inpatient) HL\r\n* No clinical evidence of disease or disease-related symptoms\r\n* A post-treatment residual mass of any size is permitted as long as it is PET negative\r\n* Spleen and liver must be non-palpable and without nodules\r\n* If a pre-treatment bone marrow biopsy was positive, an adequate bone marrow biopsy from the same site must be cleared of infiltrate; if this is indeterminate by morphology, immunohistochemistry should be negative Must consent to bone marrow aspirate or biopsy. If post allogeneic SCT must not have less than 50% donor chimerism in either peripheral blood or bone marrow Is willing to provide bone marrow biopsies and comply with protocol-defined evaluations Subject must be willing and able to undergo bone marrow aspirate per protocol (with or without bone marrow biopsy per institutional guidelines). Patients who have had RT in more than 35% of the bone marrow. Collection of a bone marrow, fluid or tissue sample that is expected to have enough cells to run the assay Failure to recover from Grade 3 or 4 toxicity from previous treatment (unrelated to malignant bone marrow involvement) All patients are required to be pre-registered to A061402 in order to submit post-radiation therapy (RT) bone marrow aspirate specimens to Roswell Park for MRD detection by flow cytometry; this submission is required prior to registration to confirm eligibility ELIGIBILITY CRITERIA - PHASE I (ARMS A, B, C): Relapsed or refractory B-cell or T-cell ALL after multi-agent chemotherapy (>= 5% marrow lymphoblasts, assessed by morphology and flow cytometry; flow cytometry will be used to confirm immunophenotype and percentage of blasts will be assessed by morphology) ELIGIBILITY CRITERIA - PHASE II (ARM D): Relapsed or refractory B-cell or T-cell ALL after multi-agent chemotherapy (>= 5% marrow lymphoblasts, assessed by morphology and flow cytometry; flow cytometry will be used to confirm immunophenotype and percentage of blasts will be assessed by morphology) Participants with > 5% involvement of bone marrow by malignant cells (either by manual count or flow cytometry) prior to stem cell collection Evidence of myelodysplasia or myeloid leukemia by morphology, immunostains, flow cytometry, or cytogenetics on a bone marrow aspirate or biopsy. Previously untreated Ph negative precursor B-cell or T-cell ALL confirmed by conventional flow cytometry or immunohistochemical stain; patients who have untreated B-cell or T-cell ALL confirmed by conventional flow cytometry or immunohistochemical stain, but Ph status is unknown, may also enroll Patients must have measurable disease requiring cytoreduction, defined as a bone marrow myeloblast count >= 5% and < 20% on morphologic examination or by flow cytometry in cases in which adequate morphologic examination is not possible Available autologous transduced peripheral blood T-cells with >= 15% expression of CAR-Kappa determined by flow-cytometry CD19 expression is required at any time since diagnosis; if patient has received anti-CD19 targeted therapy (i.e. blinatumomab), then CD19 expression must be subsequently demonstrated. CD19 expression must be detected on greater than 50% of the malignant cells by immunohistochemistry or >= 90% by flow cytometry; the choice of whether to use flow\r\ncytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each subject; in general, immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples INCLUSION - PROCUREMENT: CD5-positive tumor (result can be pending at this time); > 50% CD5 + blasts by flow cytometry or immunohistochemistry (tissue) assessed by a Clinical Laboratory Improvement Amendments (CLIA) certified flow cytometry/pathology laboratory INCLUSION - TREATMENT: CD5-positive tumor (> 50%) CD5+ blasts by flow cytometry or immunohistochemistry (tissue) assessed in a CLIA certified flow cytometry/pathology laboratory INCLUSION - TREATMENT: Available autologous activated peripheral blood T cell products with ? 20% expression of CD5.CAR.28 zeta and < 0.5% gene-modified malignant T blasts by flow cytometry TREATMENT INCLUSION: Available autologous T cells with ? 15% expression of CD30CAR determined by flow-cytometry Biopsy-confirmed CD20+ expression of the underlying malignancy by immunohistochemical staining or flow cytometry between the most recent dose of an anti-CD20 monoclonal antibody (mAb) and study enrollment Patients with any history of relapsed/refractory disease, or who have progressed at any time since beginning induction therapy are not eligible; patients who have evidence of residual disease by FISH, cytogenetics, SNP array, or flow cytometry without any measurable nodal disease or morphologic evidence of disease in the bone marrow or peripheral blood are eligible Patients will be eligible to receive donor-derived multiTAA-specific T cells following any type of allogeneic HSCT as\r\n* Adjuvant therapy for AML/MDS (Group A); or\r\n* Treatment for refractory/relapsed or minimal residual AML/MDS disease (Group B)\r\n** Residual disease at the time of transplant or post transplant relapse is defined as polymerase chain reaction (PCR) positivity, specific cytogenetic abnormalities, an abnormal population on flow cytometry or increased blasts on bone marrow biopsy, in the peripheral blood, or any other extramedullary sites\r\n* Minimal residual disease (MRD) will be defined as detection in blood, bone marrow, or other tissues any of the following:\r\n** Any leukemia specific marker such as t(8;21); inv 16; t (15;17), t(9;22) or t(4;11) documented in the patient’s leukemia cells pre-transplant on a post-transplant evaluation\r\n** Expression of a leukemia associated antigen known to be a marker for residual disease like WT1\r\n** A leukemia-specific phenotype (e.g. expression of markers including CD13 and/or CD33 and/or CD117 and/or human leukocyte antigen–antigen D related positive [HLA-DR+]) post-transplant at a level of ? 0.01%\r\n** Mixed donor chimerism (> 20%) CD22 expression must be detected on greater than 15% of the malignant cells by immunohistochemistry or greater than 80% by flow cytometry; the choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each patent; in general, immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples Patients must have measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease detected by flow cytometry, cytogenetics, or polymerase chain reaction (PCR) analysis Prior CAR therapy within 30 days prior to apheresis or prior CAR therapy at any time with evidence for persistence of CAR T cells in blood samples (circulating levels of genetically modified cells of >= 5% by flow cytometry) Available allogeneic activated peripheral blood T cell products with ? 15% expression of CD19.CAR-CD28zeta determined by flow cytometry (cell dose is based on total cell numbers and not individual anti-leukemic cell numbers) Patient’s multiple myeloma cells are positive for CD28 or CD86 expression by flow cytometry or immunohistochemistry (in any proportion) CD19 expression must be detected on greater than 15% of the malignant cells by immunohistochemistry or greater than 90% by flow cytometry; the choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each patient; in general, immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples; CD22+ B cell malignancy is required and CD22 expression levels will be documented when available, but a specific level of expression is not an eligibility requirement; it may be documented as positive or negative Prior CAR therapy within 30 days prior to apheresis or prior CAR therapy at any time with evidence for persistence of CAR T cells in blood samples (circulating levels of genetically modified cells of ? 5% in peripheral blood by flow cytometry) The patient’s lymphoma must be CD19 positive, either by immunohistochemistry or flow cytometry analysis on the last biopsy available. Evidence of CD20 expression by immunohistochemistry or flow cytometry on the tumor specimen obtained with the biopsy performed with screening CD19 expression is required at any time since diagnosis; if patient has received anti-CD19 targeted therapy (i.e. blinatumomab or CD19-CAR T cells), then CD19 expression must be subsequently demonstrated; CD19 expression must be detected on greater than 50% of the malignant cells by immunohistochemistry or >= 90% by flow cytometry; the choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each subject; in general, immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples AML Blast cell expressing CD123 by flow cytometry performed as per standard practice After CAR T cell infusion on PLAT-02, the ratio of the % CAR T cells in peripheral blood on day 10 (as measured by flow cytometry) compared to the % CAR T cells in peripheral blood on day 14 is >= 1.5\r\n* Patients meeting above criteria may enroll on Cohort B, but must demonstrate evidence of ongoing B cell aplasia (BCA) in the bone marrow within 7 days prior to planned T-APC test dose, in order to remain on Cohort B; BCA in the bone marrow is defined as < 1% CD19+ cells, as measured by flow cytometry; patients not demonstrating ongoing BCA may be considered for enrollment on Cohort C of this study Following CAR T cell infusion on PLAT-02, patient initially achieved BCA and has now lost BCA in the bone marrow or peripheral blood, before 6 months; BCA is defined as < 1% CD19+ cells by lymphocyte subset testing, as measured by flow cytometry\r\n* If patient has disease relapse in this setting, disease must remain CD19+ Evidence of CD19 expression via flow cytometry (peripheral blood or bone marrow) or immunohistochemistry (bone marrow biopsy) from a sample obtained from the current relapse Patients must have received at least 12 months of ibrutinib therapy and have measurable CLL by at least one of the following: Absolute monoclonal lymphocyte count > 4000/microL; OR measurable lymph nodes with at least one node > 1.5 cm in diameter on CT; OR bone marrow with >= 30% lymphocytes on aspirate differential; OR detectable CLL cells using a standardized flow cytometry assay for minimal residual disease Patients must have bone marrow and peripheral blood studies available for confirmation of diagnosis of AML; CD33 positivity must be confirmed by either flow cytometry or immunohistochemistry; cytogenetics, flow cytometry, and molecular studies (such as FMS-like tyrosine kinase-3 [Flt-3] status) will be obtained as per standard practice. Diagnosis of B- or T-ALL or LLy by immunophenotyping:\r\n* LLy participants must have < 25% tumor cells in bone marrow and peripheral blood by morphology and flow cytometry; if any of these show >= 25% blasts, patient will be considered to have leukemia Evidence of CD19 expression by immunohistochemistry or flow cytometry on any prior or current tumor specimen or high likelihood of CD19 expression based on disease histology CELL PROCUREMENT: CD19 positivity of lymphoblasts confirmed by flow cytometry or immunohistochemistry (IHC) per institutional standards Phase I portion of the study: Histologically or flow cytometry confirmed diagnosis of B-CLL/SLL according to National Cancer Institute (NCI)-Working Group (WG) 1996 guidelines Phase II portion of the study - histologically or flow cytometry confirmed diagnosis of BCLL/SLL according to NCI-WG 1996 guidelines; patients who lack CD23 expression on their leukemia cells should be examined for (and found NOT to have) either t(11;14) or cyclin D1 overexpression, to rule out mantle cell lymphoma Leukemia or myelodysplastic syndrome (MDS) in aplasia; these patients may be taken to transplant if after induction therapy they remain with aplastic bone marrow and no morphological or flow-cytometry evidence of disease >= 28 days post-therapy; these high risk patients will be analyzed separately Bone marrow and peripheral blood studies must be available for confirmation of diagnosis; cytogenetics, flow cytometry, and molecular studies (such as JAK-2, MPL and CALR mutational status) will be obtained as per standard practice Response to therapy and completion of at least one cycle of consolidation therapy, and with disease status meeting one of the following criterion\r\n* Minimal residual disease, as defined by detectable disease by flow cytometry but with marrow that is at least 10% cellular with < 5% blasts on morphologic review\r\n* CRi/CRp, as defined by absence of detectable disease by flow cytometry, and marrow that is at least 10% cellular with < 5% blasts on morphologic review, but with neutrophil count < 1000/ul and/or platelet count <100,000/ul. As previously stated, a platelet threshold of < 80,000/ul will be used to define CRp in pediatric patients, as per consensus pediatric response criteria\r\n* Elevated expression above of WT1 in bone marrow or peripheral blood; (increased expression of WT1 will be determined if the number of copies of WT1 divided by the number of copies of ABL x 10^4 is > 250 for bone marrow, or > 50 for peripheral blood) Minimal residual disease (MRD) positive leukemia (AML, ALL or accelerated/blast phase CML); selected patients in morphologic CR, but with positive immunophenotypic (flow cytometry) or molecular evidence of MRD may be eligible if recent chemotherapy has not resulted in MRD negative status Leukemia or MDS in aplasia; these patients may be taken to transplant if after induction therapy they remain with aplastic bone marrow and no morphological or flow-cytometry evidence of disease >= 28 days post-therapy; these high risk patients will be analyzed separately Evidence of ROR1 expression by immunohistochemistry or flow cytometry on any prior or current tumor specimen PRIOR TO LYMPHODEPLETION: Available autologous T cells with ? 15% expression of CD30CAR determined by flow cytometry required prior to treatment with ATLCAR.CD30 cells PRIOR TO INFUSION OF ATLCAR.CD30 CELLS: Available autologous T cells with ? 15% expression of CD30CAR determined by flow-cytometry required prior to treatment with ATLCAR.CD30 cells Abnormal T cells must be CD52+ as assessed by flow cytometry or immunohistochemistry Cohort 2: Persistence or reappearance of minimal residual disease by flow cytometry or cytogenetic or molecular testing while being in morphological remission after allogeneic stem cell transplantation Cohort 3: High risk AML and MDS patients who are in complete remission morphologically with no evidence of minimal residual disease by flow cytometry or cytogenetic or molecular testing after allogeneic stem cell transplantation Please note that tumor samples for patients with MF or SS can be CD30 negative and do not have to be CD30 positive on either flow cytometry or immunohistochemistry for patients to be eligible Confirmed history of CD19 positivity by flow cytometry for malignant cells AT THE TIME OF INFUSION: Available autologous transduced T lymphocytes with ? 15% expression of HER2 CAR determined by flow cytometry and killing of HER2-positive targets ? 20% in cytotoxicity assay Documented genetic lesion(s) known to confer susceptibility to inhibition by either ruxolitinib or dasatinib or cytokine receptor-like factor 2 (CRLF2) positivity by flow cytometry (for the ruxolitinib cohort) A measurable residual disease at the time of screening defined as at least MRD-positive disease:\r\n* A method of evaluation of MRD is multi-parameter flow cytometry (MFC) performed at the University of Chicago\r\n* Patients who have negative MRD by multi-parameter flow cytometry (MFC) but have residual original monoclonal protein by serum or urine immunofixation may be eligible if they are found to have MRD-positive disease by next generation sequencing (NGS) Evidence of MRD at the time of screening for this study by multi-color flow cytometry (bone marrow procedure at screening required) Histologic verification of B-cell lineage leukemia or B cell non-Hodgkin lymphoma and evidence of relapse/refractory disease with the presence of CD19 and/or CD22 by flow cytometry or immunohistochemistry of bone marrow aspirate, peripheral blood or node/tumor biopsy Evidence of marrow disease by flow and morphology after upfront or salvage cytoreductive therapy and before stem cell mobilization Patients with T cell acute lymphoblastic leukemia (ALL) must be in complete remission and minimal residual disease (MRD) negative (-) by flow cytometry and molecular studies FOR BOTH STUDY ARMS: Research participants must have bone marrow and/or peripheral blood samples available for confirmation of diagnosis of AML or BPDCN; CD123 positivity must be confirmed by either flow cytometry or immunohistochemistry within 90 days of study entry; cytogenetics, flow cytometry, and molecular studies (such as FMS-like tyrosine kinase-3 [FLT-3] status) will be obtained as per standard practice; however, for research participants who are at a high risk of recurrence, they must have historical bone marrow and/or peripheral blood samples available for confirmation of diagnosis of AML or BPDCN; CD123 positivity must be confirmed by either flow cytometry or immunohistochemistry prior to start of lymphodepletion MRD will be defined in this protocol by presence of malignant cells at 0.01% or more by flow cytometry or polymerase chain reaction (PCR) analysis at the completion of initial remission induction therapy Evidence of CD19 expression by immunohistochemistry or flow cytometry on any prior or current tumor specimen or high likelihood of CD19 expression based on disease histology Available autologous or syngeneic activated peripheral blood T cell products (CD28zeta and CD28/CD137zeta) with >= 15% expression of CD19.CAR determined by flow cytometry HLA-A2 positive based on flow cytometry History of CD19+ malignancy with evidence of relapse or persistent minimal residual disease (MRD) following autologous or allogeneic hematopoietic stem cell transplantation (cohort 1)\r\n* Relapse on this protocol is detection of CD19+ malignancies in bone marrow (>= 5%) or extramedullary lesion by morphology, cytogenetics, molecular, radiographic, and/or flow cytometry\r\n* Persistent minimal residual disease after transplantation must be demonstrated by morphology, karyotype, fluorescent in situ hybridization (FISH), flow cytometry, or reverse transcriptase (RT)-polymerase chain reaction (PCR) Measurable disease for dose expansion and lead in phases only; measurable disease defined by:\r\n* Revised International Working Group (Cheson, 2007) classification for systemic lymphoma or \r\n* Atypical and or malignant lymphocytes quantifiable by flow cytometry or morphology in blood\r\n* Or bone marrow modified severity weighted assessment tool (mSWAT) > 0 or Sezary count >= 1000 cells/uL Available autologous transduced EBV-specific cytotoxic T lymphocytes with >= 15% expression of CD30CAR determined by flow-cytometry HLA-A2 positive based on flow cytometry CD19 expression must be detected on the majority of the malignant cells by immunohistochemistry or by flow cytometry in the Laboratory of Pathology, Center for Cancer Research (CCR), National Cancer Institute (NCI), NIH; definition of which cells are malignant must be determined for each patient by the Laboratory of Pathology using techniques to demonstrate monoclonality such as kappa/lambda restriction (other techniques can be used to determine monoclonality at the discretion of the Laboratory of Pathology); the choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each patient. Immunohistochemistry will be used for lymph node biopsies and bone marrow biopsies; flow cytometry will be used for peripheral blood, fine needle aspirate, and bone marrow aspirate samples Persistent or rising minimal residual disease (MRD) after standard chemotherapy regimens: patients with evidence of minimal residual disease at the completion of therapy or evidence of rising MRD while on therapy; MRD will be defined by either flow cytometry (> 0.1% residual cells in the blast gate with immune phenotype of original leukemic clone), by molecular techniques (polymerase chain reaction [PCR] or fluorescent in situ hybridization [FISH]) or conventional cytogenetics (g-banding)\r\n* New leukemia subtypes: if new high risk features are identified after the writing of this protocol, patients can be enrolled with the approval of two members of the study committee AML or MDS relapse following allo-HSCT (morphological relapse, or MRD positive by flow cytometry, cytogenetics, molecular mutations) Confirmed history of CD19-positivity by flow cytometry for malignant cells. Expansion cohort - histologically or flow cytometry confirmed diagnosis of B-CLL/SLL according to NCI-WG 1996 guidelines; patients who lack CD23 expression on their leukemia cells should be examined for (and found NOT to have) either t(11;14) or cyclin D1 overexpression, to rule out mantle cell lymphoma Detectable clonal bone marrow plasma cells by multicolor flow cytometry and less than 10% clonal plasma cells in a bone marrow biopsy by immunohistochemistry, morphology, or flow cytometry Diagnosis should be made by bone marrow aspirate or biopsy demonstrating >= 25% involvement by lymphoblasts, with flow cytometry or immunohistochemistry confirming B-precursor or T-ALL phenotype\r\n* For patients with circulating blasts in the peripheral blood, flow cytometry confirmation of B-ALL or T-ALL phenotype is sufficient for registration onto the study; bone marrow aspirate and/or biopsy should be performed as soon as feasible, preferably prior to the initiation of any therapy\r\n* While myeloid co-expression on blasts determined to be primarily lymphoid is allowed, patients meeting World Health Organization (WHO) diagnostic criteria of mixed phenotype acute leukemia (MPAL) or leukemia of ambiguous lineage are not eligible; patients with mature B-cell phenotype are also not eligible Pre-registration: Diagnostic bone marrow and peripheral blood specimens must be submitted for eligibility testing by multiparameter flow cytometry; testing will be performed by the Eastern Cooperative Oncology Group (ECOG)-American College of Radiology Imaging Network (ACRIN) Leukemia Translational Studies Laboratory and reported to the institution Leukemic blasts must demonstrate surface expression of CD22 at the time of relapse by local/institutional flow cytometry of a bone marrow aspirate sample; (assessment of CD22 using a bright fluorophore such as phycoerythrin [PE] is strongly recommended) \r\n* In the case of an inadequate aspirate sample (dry tap) or if bone marrow aspirate is unable to be performed due to patient clinical status, flow cytometry of peripheral blood specimen may be substituted if the patient has at least 1000/uL circulating blasts; alternatively, CD22 expression may be documented by immunohistochemistry of a bone marrow biopsy specimen For acute lymphoblastic leukemia: diagnosis should be made by bone marrow aspirate or biopsy demonstrating >= 30% involvement by lymphoblasts, with flow cytometry or immunohistochemistry confirming B-precursor or T-ALL phenotype; while myeloid co-expression on blasts determined to be primarily lymphoid is allowed, patients with leukemia of ambiguous lineage are not eligible For patients with circulating blasts in the peripheral blood, flow cytometry confirmation of B-precursor or T-ALL phenotype is sufficient for registration onto the study; bone marrow aspirate and/or biopsy should be performed as soon as feasible, preferably prior to the initiation of any therapy Recipient leukocyte infusion (RLI) might involve the infusion of circulating tumor cells to the patients; to minimize this risk, patients who have evidence of circulating tumor cells by light microscopy and flow cytometry will be excluded Patients will NOT receive a second infusion of GD-2 CAR T cells if >= 5 % of the circulating T cells are GD2-CAR positive by flow cytometry Bone marrow and peripheral blood studies must be available for confirmation of diagnosis; cytogenetics, flow cytometry, and molecular studies (such as Flt-3 status) will be obtained as per standard practice CD20+ bone marrow or lymph node by immunohistochemistry or flow cytometry\n obtained within 28 days prior to registration CD19 expression must be detected on greater than 15% of the malignant cells by immunohistochemistry or greater than 30% by flow cytometry in a Clinical Laboratory Improvement Amendments (CLIA) approved test in the Laboratory of Pathology, Center for Cancer Research (CCR), National Cancer Institute (NCI), National Institutes of Health (NIH) or from the referring institution or reference laboratory; the choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each patient; in general immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples Patients must have measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease detected by flow cytometry, cytogenetics, or polymerase chain reaction (PCR) analysis Patients with evidence of myelodysplasia, leukemia by morphology, immunostains flow cytometry or abnormal cytogenetics on a bone- marrow aspirate or biopsy; the diagnosis of myelodysplasia will be made by an independent investigator of the Laboratory of Pathology, National Cancer Institute (NCI) taking into consideration the totality of the clinical, pathological, flow cytometric and cytogenetic and present in a particular individual’s evaluation Patients will be tested for human leukocyte antigen A0201 (HLA-A0201) as determined by flow cytometry followed by molecular analysis of a peripheral blood specimen; however this result will not be an inclusion criterion Available autologous transduced EBV-specific cytotoxic T lymphocytes with >= 15% expression of HER2 CAR determined by flow-cytometry and killing of HER2-positive targets >= 20% in cytotoxicity assay Presence of circulating lymphoma cells by morphology or flow cytometry (> 0.1%) at or near the time of peripheral blood stem cell (PBSC) collection if unpurged/unselected PBSC are to be used (patients with cryopreserved stem cells which are negative [=< 0.1% involved] by flow cytometry will also be considered eligible) Available autologous transduced peripheral blood T-cells with >= 15% expression of CD19CAR determined by flow-cytometry Bone marrow aspirate or biopsy must have ? 5% blasts by morphology and/or flow cytometry. For relapsed patients, CD19 tumor expression demonstrated in bone marrow or peripheral blood by flow cytometry within 3 months of study entry For relapsed patients, documentation of CD19 tumor expression demonstrated in bone marrow or peripheral blood by flow cytometry within 3 months of study entry. PNH diagnosis confirmed by documented by high-sensitivity flow cytometry Evidence of myelodysplasia or myeloid leukemia by morphology, immunostains, flow cytometry, or cytogenetics on a bone marrow aspirate or biopsy. Patients must have a history of relapsed/refractory CD19+ B-ALL involving the marrow to be eligible for infusion of modified T cells; please note >= 5% blasts by morphology, fluorescent in situ hybridization (FISH)/cytogenetics, molecular translocation and/or flow cytometry constitutes a bone marrow relapse on this protocol \r\n* Patients must also fulfill one of the following criteria to be eligible for infusion of modified T cells:\r\n** Second or greater (>= 2) relapse\r\n** Early first marrow relapse (1st CR < 18 months)\r\n** Intermediate/late first marrow relapse (1st CR >= 18 months from 1st CR) with poor initial response (>= 5% blasts by morphology and/or flow cytometry) following re-induction chemotherapy\r\n** Refractory disease\r\n** Ineligible for hematopoietic stem cell transplant (HSCT) as determined by the treating physician in consultation with the bone marrow transplant (BMT) service\r\n** Patient would not benefit from additional cytotoxic chemotherapy as determined by the treating physician PNH diagnosis confirmed by documented by high-sensitivity flow cytometry. Neoplastic mast cells must express CD30 by immunohistochemistry or flow cytometry Bone marrow and peripheral blood studies must be available for confirmation of diagnosis; cytogenetics, flow cytometry, and molecular studies (such as Fms-like tyrosine kinase 3 [FLT-3] status) will be obtained as per standard practice Available autologous T cells with >= 15% expression of CD30CAR determined by flow-cytometry Measurable disease defined by: \r\n* Lugano classification for systemic lymphoma or\r\n* Atypical and or malignant lymphocytes quantifiable by flow cytometry or morphology in blood or bone marrow or \r\n* Modified severity-weighted assessment tool (mSWAT) > 0 or Sezary count >= 1000 cells/uL for CTCL Have documented CD45 expression by leukemic cells via flow cytometry (a \blast gate\ on CD45 vs. side scatter analysis consistent with AML) Diagnosis of PNH by flow cytometry Additionally, patients who were previously MRD negative for >= 3 months after induction therapy with/without consolidative HDT/ASCT and have turned MRD positive (by flow cytometry) within the last 3 months and do not have any evidence of progressive disease are eligible. Untreated, histological confirmed acute myeloid leukemia (AML) based on World Health Organization (WHO) 2008 criteria with Kit expression (CD 117) of myeloblasts >= 20% by flow cytometry from bone marrow aspirate at diagnosis Patient must have > 10% disease burden measured by cytomorphology, flow cytometry, or cytogenetics Individuals whose lymphoblasts have surface immunoglobulin by flow cytometry and/or known t(8;14), t(2;8), or t(8;22); absence of surface immunoglobulin by flow cytometry at time of initial diagnosis or relapse is sufficient to rule out mature B-cell leukemia; karyotype or fluorescent in situ hybridization (FISH) results documenting absence of t(8;14), t(2;8), or t(8;22) are not necessary prior to enrollment if absence of surface of immunoglobulin is documented by flow cytometry At least 1 of the following high-risk features for previously untreated patients:\r\n* Rai stage II disease\r\n* Rai stage 0-I with disease-related fatigue\r\n* Serum beta 2 microglobulin (beta2M) >= 3 mg/L\r\n* Absolute lymphocyte count >= 25,000/uL\r\n* Unmutated immunoglobulin heavy variable cluster (IGHV) gene or IGHV3-21\r\n* Zeta-chain-associated protein kinase 70kDa (ZAP70) positive (>= 20% by flow cytometry or positive by immunohistochemistry)\r\n* CD38 positive (>= 30% by flow cytometry); OR\r\n* Deletion 11q or 17p by fluorescent in situ hybridization (FISH) Tissue diagnosis of lymphoplasmacytic lymphoma with surface immunoglobulin G (IgG), immunoglobulin A (IgA) or immunoglobulin M (IgM) phenotype with a monoclonal heavy and light chain as determined by flow cytometry; all primary diagnostic lymph node and/or bone marrow biopsies will be reviewed at the University of Texas M.D. Anderson Cancer Center (UTMDACC) Patients must have a diagnosis of B-ALL by flow cytometry, bone marrow histology, and/or cytogenetics Patients must have CD19+ acute lymphoblastic leukemia (ALL) as confirmed by flow cytometry and/or immunohistochemistry Diagnosis of CLL by immunophenotyping and flow cytometry analysis of blood or bone marrow. Confirmed B-cell CLL/SLL with a characteristic immunophenotype by flow cytometry, and symptomatic disease requiring treatment Confirmation of diagnosis of B cell malignancy and positivity for CD19 confirmed by the Laboratory of Pathology of the National Cancer Institute (NCI); the choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each patient; immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood, fine needle aspirates and bone marrow samples With minimal residual disease (MRD) or relapse post-HSCT (for the phase I dose escalation) as evidenced by polymerase chain reaction (PCR) positivity, specific cytogenetic abnormalities, an abnormal population on flow cytometry or increased blasts on bone marrow biopsy or in the peripheral blood Available allogeneic CD19CAR transduced tri-virus-specific cytotoxic T lymphocytes with >=15% expression of CD19CAR determined by flow cytometry and greater than 10% killing of one or more viral antigen pulsed targets in a cytotoxicity assay at an effector:target ratio of 20:1 Evidence of bone marrow MRD defined as ? 0.01% by flow cytometry performed in the study central lab Patients with leukemic form of PTCL who will not have a measurable lesion in two dimensions by CT scan, relapsed or refractory disease must be detected by immunohistochemistry or flow cytometry and molecular clonality studies in bone marrow or peripheral blood Greater than 20% aberrant intraepithelial lymphocytes (IEL) as assessed by flow cytometry AML blasts must express CD30 (>= 10% expression as assessed by flow-cytometry or 2+ expression by immunohistochemistry) (whenever possible CD30 expression will be assessed by both methods) Histological diagnosis of Diffuse Large B Cell Lymphoma (de novo or transformed) expressing CD19 by immunohistochemistry or flow cytometry analysis (>30% positivity), based on recent (less than 6 months) or new biopsy. Presence of hepatopetal flow Hematologic malignancy in complete remission with minimal residual disease (MRD) detectable by conventional cytogenetics, fluorescence in situ hybridization (FISH), flow cytometry, or molecular testing Bone marrow and peripheral blood studies must be available for confirmation of diagnosis; cytogenetics, flow cytometry, and molecular studies (such as JAK-2, myeloproliferative leukemia [MPL] and calreticulin [CALR] mutational status) will be obtained as per standard practice Available autologous T cells with >= 15% expression of CD30CAR determined by flow-cytometry required prior to treatment with ATLCAR.CD30 cells The evidence of CD19+ expression on leukemia cells must be confirmed by pathology review of the bone marrow and/or peripheral blood specimens (flow cytometry and/or immunohistochemistry) collected at the time of current relapse and prior to the initiation of therapy Criteria for Solid Tumor Expansion and Lymphoma Cohorts:\r\n* Inclusion and exclusion criteria for this cohort are the same as above, with the following rule for CD4 count based on tolerability in Phase I; if, participants with lymphocyte T CD4 count between 100-200/mm^3 (Stratum 2) are shown to tolerate treatment in the Phase I dose de-escalation portion at the same dose level as those with CD4 counts > 200/mm^3 (Stratum 1), participants in the expansion cohort with CD4 counts >= 100/mm^3 are permitted; otherwise, the expansion is open to all solid tumor patients except those whose tumors are known not to respond to nivolumab (pancreas, prostate and MSS colon cancer); for the relapsed refractory HIV-cHL cohort, participants with CD4 count >= 100/mm^3 are permitted CD4 count >= 50 cells/ul Patients who have BCR-ABL fusion transcript determined by fluorescence in situ hybridization (FISH) or real time-polymerase chain reaction (RT-PCR) or t(9;22)(q34;q11) by cytogenetics are not eligible and should be considered for enrollment on studies that incorporate imatinib during induction; please note: patients must also be assessed for CD20 positivity and other markers; positivity for CD22 and CD20 is defined as baseline expression of the CD22 or CD20 antigen in more than 20% of leukemic cells using local multiparameter flow-cytometric immunophenotyping with the use of CD45 expression as a marker to gate the ALL blast population, according to recommendations from the European LeukemiaNet Participants must have a CD4 count performed within 30 days of enrollment ELIGIBILITY CRITERIA - PHASE II (ARM D): Patient cannot have poorly controlled HIV, or CD4 < 400; HIV positive patients are allowed on this study if they have a CD4 count >= 400, and are on a stable antiviral regimen Dose Expansion Cohort #1 - Patients will have relapse of CD123+ BPDCN. Patients with prior CD123-targeting agents will be allowed as long as the blasts still have detectable CD123 expression. Histologically documented CD20-positive lymphoma Known CD20-negative status at relapse or progression Prior treatment with CD47 or signal regulatory protein alpha (SIRP?) targeting agents. Clinical and phenotypic verification of B cell CLL/ SLL/ or monoclonal B-cell lymphocytosis (MBL) and measurable disease; immunophenotyping of the leukemic cells (blood, lymph node, or bone marrow) must demonstrate a monoclonal (kappa or lambda light chain restricted) B cell population with immunophenotype diagnostic for CLL (e.g.: co-expressing CD19 and CD5) Histologically documented CD20-positive B-cell lymphoma as determined by the local laboratory Known CD20-negative status at relapse or progression Prior treatment with the following agents (from last dose of prior treatment to first dose of GSK3174998): Tumor necrosis factor receptor (TNFR) agonists, including OX40, CD27, CD137 (4-1BB), CD357 (GITR): at any time; Checkpoint inhibitors, including PD-1, PD-L1, and anti-cytotoxic T-lymphocyte antigen 4 (CTLA-4) inhibitors: within 4 weeks. ADJUVANT COHORT: Derived benefit from PD 0332991 in the neoadjuvant setting in this trial; this includes the 26 patients who achieved complete cell cycle arrest only after the addition of PD 0332991 (C1D1 Ki67 > 2.7% and C1D15 Ki67 =< 2.7%) from the main study (PIK3CA WT, mutant, or unknown cohorts) as well as any patients who have a Ki67 =< 10% on C1D15 biopsy in the endocrine resistant cohort Patient with C-kit (CD117) positive tumour detected immuno-histochemically Obtained within 14 days prior to C1D1: HCV NA analysis negative Subjects who have undergone prior anti-CD19 or anti-CD22 CAR therapy will be eligible if < 5% of circulating levels of CD3+ cells express the previous CAR by flow cytometry >= 1% of CD30-positivity by immunohistochemistry confirmed by hematopathology review at the participating institution Prior treatment with a CD47 or signal regulatory protein (SIRP) alpha targeting agent. Must have received adequate prior therapy for the underlying CD20+ B-cell lymphoma, defined as an anti-CD20 mAb in combination with an anthracycline-containing chemotherapy regimen (i.e. chemo-immunotherapy) and at least one of the following: Known CD20-negative status at relapse or progression Residual disease at the time of transplant or post-transplant relapse is defined as polymerase chain reaction (PCR) positivity, specific cytogenetic abnormalities, an abnormal population on flow cytometry, or increased blasts on bone marrow biopsy or in the peripheral blood; minimal residual disease (MRD) will be defined as detection in blood or marrow of any of the following:\r\n* Any leukemia-specific marker (such as t(12;21); t(9;22) or t(4;11)) documented in the patient’s leukemia cells pre-transplant on a post-transplant evaluation\r\n* A leukemia-specific phenotype (e.g. expression of markers including CD10 and/or CD19 or CD3 and/or CD4 or CD8) post-transplant at a level of ? 0.01%\r\n* Mixed donor chimerism (> 20%) Presence of lymphadenopathy and/or splenomegaly with histopathological evaluation of a lymph node biopsy consistent with CLL. Clonality of the circulating B-lymphocytes should be confirmed at screening. Previously confirmed immunohistological diagnosis with a characteristic CD5+/CD20+ B--cell immunophenotype according to WHO criteria. CLL diagnosis requires an absolute peripheral blood monoclonal CD20+/CD5+ B-lymphocyte count ? 5000 cells/?L for the duration of at least 3 months. Part 2: Diagnosis of CLL according to the National Cancer Institute (NCI) criteria or SLL according to the World Health Organization (WHO) criteria; this includes previous documentation of:\r\n* Biopsy-proven small lymphocytic lymphoma or\r\n* Diagnosis of CLL according to NCI working group criteria as evidenced by all of the following:\r\n** Peripheral blood B cell count of > 5 x 10^9/L consisting of small to moderate size lymphocytes\r\n** Immunophenotyping consistent with CLL defined as:\r\n*** The predominant population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20 [typically dim expression] or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc.)\r\n*** Clonality as evidenced by kappa or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. immunoglobulin heavy chain variable [IGHV] analysis)\r\n*** NOTE: splenomegaly, hepatomegaly, or lymphadenopathy are not required for the diagnosis of CLL\r\n** Before diagnosing CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescent in situ hybridization (FISH) analysis for t(11;14)(immunoglobulin H [IgH]/cyclin D1 [CCND1]) on peripheral blood or tissue biopsy or negative immunohistochemical stains for cyclin D1 on involved tissue biopsy Clinical and phenotypic verification of B cell CLL and measurable disease; immunophenotyping of the leukemic cells (blood or marrow) must demonstrate a monoclonal (or light chain positive) with immunophenotype consistent with B cell population (e.g., co-expressing cluster of differentiation [CD]19 and CD5) CD19+ leukemia RETREATMENT WITH MODIFIED T-CELLS INCLUSION CRITERIA: Subject has evidence of persistence of leukemic cells OR has CD19+ B cell recovery detected within 1 year of initial T cell infusion Patients must have measurable or evaluable disease; patients with > 10% of the peripheral blood mononuclear cells (PBMCs) having the characteristic abnormal (i.e., cluster of differentiation [CD]3dim, CD4+ CD25+ expressing) fluorescence-activated cell sorting (FACS) profile for circulating ATL cells will be considered to have evaluable disease Patients must have histologically confirmed B-lineage acute lymphoblastic leukemia (ALL) at diagnosis and either evidence of relapse/refractory disease based on a bone marrow/peripheral blood examination or evidence by cytogenetic studies or polymerase chain reaction (PCR) amplification; patients with only extramedullary disease in the absence of bone marrow or blood involvement are not eligible; patients with L3 (Burkitt's) are not eligible; for ALL in marrow or peripheral blood, immunophenotyping of the blood or marrow lymphoblasts must be performed to determine lineage (B cell, T-cell, or mixed B/T cell); NOTE: appropriate marker studies including CD19 (B cell), CD10, CD5, and CD7 (T cell) must be performed; co-expression of myeloid antigens (CD13 and CD33) will not exclude patients; if possible, the lineage specific markers cytoplasmic CD22 or CD79a (B cells), cytoplasmic CD3 (T cells) and cytoplasmic myeloperoxidase (MPO) (myeloid cells) must be determined; patients with mixed lineage ALL (ML-ALL) as defined by a lack of cytochemical markers of myeloid differentiation, and by the presence of immunophenotypic markers suggesting both lymphoid and myeloid differentiation, are allowed CD19 and/or CD22 must be expressed on at least 50% of the lymphoblasts Subjects must be diagnosed with CLL/SLL (chronic lymphocytic leukemia/small lymphocytic lymphoma) based on the standard histologic and immunophenotypic criteria described in the World Health Organization (WHO) classification of lymphoid malignancies, including immunophenotypic confirmation that the tumor cells co-express B cell antigens cluster of differentiation (CD)19/20 and CD5; mantle cell lymphoma should be excluded based on positive staining of the tumor cells for CD23, or the absence of staining of the tumor cells for cyclin D1 or the absence of t(11;14); this diagnosis should be confirmed at a Dana-Farber Harvard Cancer Center institution (Dana-Farber Cancer Institute [DFCI], Brigham and Women's Hospital [BWH], Massachusetts General Hospital [MGH], Beth Israel Deaconess Medical Center [BIDMC]) within approximately one month after the subject is registered; any question on histology confirmation should be brought to the attention of the Principal Investigator Evidence of HCL by flow cytometry, reviewed by the Laboratory of Pathology, National Cancer Institute (NCI), including positivity for CD19, CD22, CD20, and CD11c TREATMENT WITH SJCAR19: Available SJCAR19 product with >= 15% expression of the CD19-chimeric antigen receptor (CAR), and killing of CD19+ targets >= 20% in an in vitro cytotoxicity assay Prior treatment with a therapeutic agent targeting CD33 and/or CD3 (e.g. gemtuzumab ozogamicin, SGN-CD33A or AMG 330). Radiotherapy within the last 21 days prior to C1D1 (limited field palliative radiation is allowed if ? 14 days prior to C1D1); Prior treatment with CD74 targeting therapy Clinical and phenotypic verification of B cell CLL or SLL and measurable disease. Immunophenotyping of the leukemic cells (blood, lymph node, or bone marrow) must demonstrate a monoclonal (or light chain positive) B cell population with immunophenotype consistent with CLL (e.g.: co-expressing CD19, CD5, and CD23) Patients with HCL variant (as defined by absence of expression of CD25) Prior CD19 directed therapy other than blinatumomab Prior treatment with blinatumomab or CD-directed CAR T-cell therapy Evidence of at least one measurable lesion on imaging with the following exceptions\r\n* Patients treated with interim chemotherapy for disease control between enrollment and ET190L1-ARTEMIS T cell infusion who do not have measurable disease at re-screening are still eligible\r\n* CLL/SLL with documented B-cell absolute lymphocytosis > 5 x 10^9 cells/L peripheral blood or infiltration of lymph nodes and/or bone marrow infiltration by CLL phenotype cells defined as clonal B cells with majority population co-expressing CD5 and CD19, with surface immunoglobulin (sIg, kappa or lambda but not both) and CD20 (dim), CD23+ (if CD20 or sIg are bright or if CD23 is dim or negative [atypical CLL phenotype] then fluorescence in situ hybridization (FISH) for 11:14 translocation must be performed to differentiate from mantle cell lymphoma)\r\n* Lesions previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy\r\n** Measurable lesions are nodes/nodal masses > 1.5 cm, extranodal masses > 1.0 cm or positron emission tomography (PET) avid lesions consistent with lymphoma CD19+22+ leukemia Subjects must NOT have an active malignancy other than CD19+CD22+ leukemia CD4 count > 200/uL Previous treatment with CD19-targeted CAR T cells Subjects who have undergone prior anti-CD19 or anti-CD22 CAR therapy will be eligible if < 5% of circulating levels of CD3+ cells express the previous CAR by flow cytometry Documentation of CD20+ status Histologically confirmed CD30-positive lymphoma with cutaneous presentation biopsy-confirmed CD20+ expression of the underlying malignancy with disease progression following immediate prior therapy must have received adequate prior therapy for the underlying CD20+ B-cell lymphoma, defined as an anti-CD20 mAb in combination with an anthracycline-containing chemotherapy regimen (i.e. chemo-immunotherapy) and at least one of the following: Prior to lymphodepletion on protocol PLAT-02, total CD19 antigen load in the bone marrow is <15% of cells analyzed by flow cytometry (CD19 antigen load includes both CD19+ leukemia cells and non-malignant B cells) Patients not in remission must have CD45-expressing leukemic blasts; patients in remission do not require phenotyping and may have leukemia previously documented to be CD45 negative Documentation of CD19 expression on any prior or current tumor biopsy; patients who have received previous CD19-targeted therapy must have CD19-positive disease confirmed on a biopsy since completing the prior CD19-targeted therapy CRITERIA FOR SCREENING: For patients in stage 1 only, prior treatment with any CD19 CAR T-cell therapy is excluded Clear CD30 expression must be detected on 75% or more of malignant cells from either bone marrow or lymphoma mass by flow cytometry or immunohistochemistry; the patient’s malignancy will need to be assessed for CD30 expression by flow cytometry or immunohistochemistry performed at the National Institutes of Health (NIH); if unstained, paraffin-embedded bone marrow or lymphoma sections are available from prior biopsies, these can be used to determine CD30 expression by immunohistochemistry; otherwise, patients will need to come to the NIH for a biopsy to determine CD30 expression; the sample for CD30 expression can come from a biopsy obtained at any time before enrollment, unless the patient has received a prior anti-CD30 monoclonal antibody, in which case the sample must come from a biopsy following completion of the most recent anti-CD30 monoclonal antibody treatment Documented CD20+ FL. CD4 count >= 100 in the dose-finding cohort; once the dose-finding cohort is complete and if safety is established, participants with any CD4 count, including CD4 count < 100, will be allowed in the dose-escalation phase Participants who have received previous CD19-targeted therapy must have CD19-positive lymphoma confirmed on a biopsy since completing the prior CD19-targeted therapy. Pathologically confirmed MCL, with documentation of monoclonal CD20+ B cells that have a chromosome translocation t(11;14)(q13;q32) and/or overexpress cyclin D1. Subjects who had a thromboembolic event ? 4 weeks of C1D1 must be receiving adequate anticoagulation treatment for at least 2 weeks before C1D1 and must continue as clinically indicated post first dose Positive CD30+ immunohistochemical expression Sufficient baseline tumor tissue available to perform tumor infiltrating CD8+ T-cell density assessment; (NOTE: The actual CD8+ T-cell density assessment does not need to be performed prior to registration, however the slides must be reviewed prior to registration to ensure that adequate tissue will be available for the pre-treatment tumor infiltrating CD8+ T-cell density assessment) Measurable unresectable cancer expressing CD70 as assessed by immunohistochemistry of resected tissue (>= 2+ CD70 positive on >= 50% of cancer cells, or >= 1+ CD70 positive on >= 75% of cancer cells) ELIGIBILITY FOR TREATMENT WITH TCRC4-TRANSDUCED CD8+ CELLS EXCLUSION FOR TREATMENT WITH TCRC4-TRANSDUCED CD8+ CELLS CD19 negative B-cell leukaemia Subjects must have a diagnosis of relapsed or refractory mantle cell lymphoma as follows:\r\n* Diagnosis of mantle cell lymphoma (MCL) must include morphology and expression of either cyclin D1 in association with other relevant markers (eg, CD19, CD20, CD5) or evidence of t(11;14) as assessed by cytogenetics, fluorescent in situ hybridization (FISH), or polymerase chain reaction (PCR)\r\n* Relapsed or refractory disease is defined as no response or progressive disease to prior treatment if the prior treatment comprised any of the following:\r\n** 1 regimen containing an anti-CD20 antibody administered for >= 2 doses, and/or\r\n** >= 1 regimen containing 1 cytotoxic agent (e.g., bendamustine, chlorambucil, cyclophosphamide, cytarabine, doxorubicin) administered for 2 cycles Prior treatment with any CD19 CAR T-cell therapy Prior treatment with programmed cell death (PD)-1, PD-ligand (L)1, cytotoxic T lymphocyte-associated protein 4 (CTLA 4) targeted therapy, or tumor necrosis factor receptor superfamily (TNFRSF) agonists including CD134 (OX40), CD27, CD137 (4-1BB), and CD357 (glucocorticoid-induced tumor necrosis factor receptor family-related protein [GITR]) Evidence of CD19 expression on any prior or current tumor specimen or a high likelihood of CD19 expression based on disease histology Documentation of CD19 expression on any prior or current tumor biopsy PRIOR TO CELL PROCUREMENT: CD30+ disease (result can be pending at the time of cell procurement, but must be confirmed prior to treatment with ATLCAR.CD30 cells); NOTE: CD30+ disease requires documented CD30 expression by immunohistochemistry based on the institutional hematopathology standard PRIOR TO LYMPHODEPLETION: CD30+ disease (result can be pending at the time of cell procurement, but must be confirmed prior to lymphodepletion); NOTE: CD30+ disease requires documented CD30 expression by immunohistochemistry based on the institutional hematopathology standard PRIOR TO INFUSION OF ATLCAR.CD30 CELLS: AST ? 3 times ULN PRIOR TO INFUSION OF ATLCAR.CD30 CELLS: Serum creatinine ? 1.5 times ULN Patients must have measurable or evaluable disease; NOTE: All patients with greater than 10% abnormal CD4+ homogeneous CD3^low strongly CD25+ expressing cells, or greater than 5% Sezary/T-PLL cells, among the peripheral blood mononuclear cells (PBMCs) in the peripheral blood will be deemed to have evaluable disease CD4 count >= 100 cells/uL if HIV-infected Exposure to another CD37 targeting drug. Documentation of CD22 expression on malignant cells at relapse PATIENT CRITERIA FOR PROCEEDING WITH CD8+ MEMORY T-CELL INFUSION: Patients must have evidence of mixed CD3 T-cell chimerism based on the day +28 (+/- 7 days) blood sample showing >= 30% and =< 90% donor type cells Diagnosis of CLL: monoclonal B-cells co-expressing >= one B-cell marker (cluster of differentiation [CD]19, CD20, or CD23) and CD5 in peripheral blood or lymph node Expression of CD-22 in >= 20% blasts DONOR: Cell yield goal (post CD34 determination): > 1-6 x10^6 CD34+ cells/kg recipient CD19-positive tumor Patients should have a confirmed diagnosis of chronic lymphocytic leukemia defined as all of the following:\r\n* Absolute lymphocyte count (ALC) > 5000\r\n* Positive for either CD19 or CD 20 together with CD23 and CD5\r\n* Less than 55% atypical cells Documented expression of CD30 on tumor cells Patients must have histological confirmation of the diagnosis (it is recommended that the immunohistochemical panel includes: CD45, CD20, CD30, CD15, CD10, BCL6, BCL2, MUM-1), and in addition have a dominant mass within the anterior mediastinum. Previous infusion of CD19 CAR T cells at another institution CD30 positive tumor (can be pending at this time) Diagnosis - CD30+ HL or CD30+ NHL After Dose Escalation: any patient (children or adults) with relapsed CD30+ HL or CD30+ NHL or newly diagnosed patients unable to receive or complete standard therapy OR diagnosis of relapsed/refractory CD30+ HL or CD30+ NHL with a treatment plan that will include high dose therapy and autologous stem cell transplantation CD30 positive tumor Evidence of HCL by flow cytometry of blood or a solid (lymph node) mass, confirmed by the Laboratory of Pathology, National Cancer Institute (NCI), including positivity for CD19, CD22, CD20, and CD11c; patients with flow cytometry consistent with HCL variant (HCLv) are eligible, including those with CD25 and/or CD103 negative disease Prior treatment with the following agents: a) Tumor necrosis factor receptor (TNFR) agonists, including OX40, cluster of differentiation (CD)27, CD137 (4-1BB), CD357 (glucocorticoid-induced TNFR family-related gene) at any time. b) TLR4 agonist at any time. c) Anticancer therapy or investigational therapy within 30 days or 5 half-lives of the drug, whichever is shorter. d) Prior radiation therapy: permissible if at least 1 non-irradiated measurable lesion is available for assessment according to RECIST version 1.1 or if a solitary measurable lesion was irradiated, objective progression is documented. A wash out of at least 14 days before start of study treatment for radiation of any intended use to the extremities for bone metastases and 28 days for radiation to the chest, brain, or visceral organs is required. Any prior exposure to Hu5F9?G4 or other CD47 targeting agents. Patients must have a confirmed diagnosis of CLL as defined by the International Workshop on CLL 2008 (iwCLL2008) criteria below:\r\n* Presence of at least 5 x 10^9 B lymphocytes/L (5000/uL) in the peripheral blood\r\n* Morphologically, the lymphocytes must appear of small to moderate size with < 55% prolymphocytes, atypical lymphocytes or lymphoblasts\r\n* The clonality and immunophenotype of the circulating B-lymphocytes must be confirmed by flow cytometry to express cluster of differentiation (CD)5, CD23, CD19, CD20, CD52 and either kappa or lambda light chain Diagnosis of:\r\n* Biopsy-proven small lymphocytic lymphoma (SLL) , or\r\n* Diagnosis of chronic lymphocytic leukemia (CLL) with a clonal B-cell population in the peripheral blood with immunophenotyping consistent with CLL as follows:\r\n** The population of lymphocytes share both B-cell antigens (CD19, CD20 [typically dim expression], or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc.)\r\n** Clonality as evidenced by kappa or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. IGHV analysis)\r\n** Before diagnosing CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescence in situ hybridization (FISH) analysis for t(11;14)(IgH/CCND1) CD30-positivity by immunohistochemistry of >= 1% Confirmed diagnosis of MCL with CD20 and cyclin D1 positivity in tissue biopsy. Prior treatment with CD47 or signal regulatory protein alpha (SIRP?) targeting agents Patients must have newly diagnosed T-lymphoblastic leukemia (T-ALL) or T-lymphoblastic lymphoma (T-LLy) stages II-IV \r\n* Note: a diagnosis of T-ALL is established when leukemic blasts lack myeloperoxidase or evidence of B-lineage derivation (cluster of differentiation [CD]19/CD22/CD20), and express either surface or cytoplasmic CD3 or two or more of the antigens CD8, CD7, CD5, CD4, CD2 or CD1a, and are present either in peripheral blood or > 25% in the bone marrow; if surface CD3 is expressed on all leukemic cells, additional markers of immaturity, including terminal deoxynucleotidyl transferase (TdT), CD34 or CD99 will be assessed for expression; cases with uncertain expression will receive additional review within the appropriate Children's Oncology Group (COG) reference laboratory\r\n* For T-LLy patients with tissue available for flow cytometry, the criterion for diagnosis should be analogous to T-ALL; for tissue processed by other means (i.e. paraffin blocks), the methodology and criteria for immunophenotypic analysis to establish the diagnosis of T-LLy defined by the submitting institution will be accepted Patients must have histologically or flow cytometry confirmed diagnosis of B-cell chronic lymphocytic leukemia/small lymphocytic lymphoma (B-CLL/SLL) according to the IWCLL 2008 criteria. The malignant B cells must co-express CD5 with CD19 or CD20. Patients who lack CD23 expression on their leukemia cells should be examined for (and found NOT to have) either t(11;14) or cyclin D1 overexpression, to rule out mantle cell lymphoma. Severe CD as defined by one of the following:\r\n* CDAI >= 250\r\n* Need for total parenteral nutrition to maintain weight\r\n* Recurrent intestinal inflammation caused by CD following surgical resection Subjects must complete the filgrastim (G-CSF)/plerixafor mobilization of peripheral blood progenitor cells and must have collected at least 7.5 x 10^6 CD34+ cells/kg, sufficient for preparation of both, a back-up of 2.5 x 10^6 CD34+ cells and the research product Diagnosis of CD-30 positive GCT; CD30 expression will be tested by immunohistochemistry (IHC) in archival or fresh tumor tissue as is routinely done for diagnosis At least 10% of the cells obtained from lymph node, or extranodal sites must react with anti-CD25 (anti-Tac) on immunofluorescent or immunoperoxidase staining; patients with CD25-positive infiltrating T cells will be eligible even if their Hodgkin’s (Reed-Sternberg) cells are CD25-negative Autologous graft with a minimum of >= 3.0 x 10^6 CD34+ cells/kg; not CD34 selected Patients must have a histologically confirmed diagnosis of B-NHL, T-NHL, or HL; CD45 antigen expression must be documented on tumor specimens in all cases except HL, in whom histologic demonstration of CD45+ cells adjacent to the Reed Sternberg cells is required Histopathological evidence of CD25+ HCL confirmed by the National Institutes of Health (NIH) pathology department; this will require a monoclonal population of peripheral malignant lymphocytes that are CD25 positive by fluorescence activated cell sorting (FACS) with anti-CD25 antibody; positive expression in a FACS assay is defined as more than 2 times the mean fluorescence intensity (MFI) of the control antibody by FACS; HCLv (HCL variant) is usually CD25 negative, and eligibility would require CD25+ HCLv Prior treatment with CD19 directed agents unless CD19 expression is confirmed after completion of CD19-directed treatment The lymphoma must express the cluster of differentiation (CD)20 antigen by either flow cytometry using anti-CD20 antibodies or by immunoperoxidase staining of paraffin sections; a report providing confirmation of CD20 expression must be submitted Patients who have an option for any treatment with proven clinical benefit for CD25-positive AML or CD25-positive ALL at current state of disease. CD4+ T-cell count >= 100 cells/uL Expression of CD30 Histologically documented CD20-positive lymphoma as determined by the local laboratory Known CD20-negative status at relapse or progression Patients must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective; previously treated, pathologically confirmed chronic lymphocytic leukemia (CLL) or small lymphocytic lymphoma (SLL) that requires as per the National Cancer Institute (NCI) Working Group Guidelines for the treatment of CLL; the diagnosis of CLL is defined by the presence of 5 x 10^9 clonal B-lymphocytes/L in the peripheral blood; clonality of the lymphocyte population is established with flow cytometry and the demonstration of the following surface markers: CD5, CD23, CD19, CD20 and the presence of either kappa or lambda immunoglobulins; the diagnosis of SLL may be made with the demonstration of < 5 x 10^9 clonal B-lymphocytes/L in the peripheral blood, with the clinical or radiographic features of enlarged lymph nodes or organomegaly, and the demonstration of SLL cells in the lymph node biopsy; institutional flow cytometry or immunohistochemistry must confirm CD20 antigen expression; patients may not have had a history of Richter’s transformation CD30 immunohistochemical staining using the anti-CD30 Becton Dickinson monoclonal (BerH2) antibody must be available on the most recent biopsy specimen; during dose escalation, patients can be either CD30 positive or CD30 negative; during dose expansion, 15 patients must be CD30 positive and 15 patients must be CD30 negative Immunotherapy within 6 months of C1D1 Available CD34+ stem cells Prior treatment with a therapeutic agent targeting CD19 and/or CD3 Patients must be diagnosed with CLL in accordance with International Workshop on Chronic Lymphocytic Leukemia (IWCLL) 2008 criteria that includes all of the following:\r\n* >= 5 x 10^9 B lymphocytes (5000/uL) in the peripheral blood\r\n* On morphologic review, the leukemic cells must be small mature lymphocytes, and prolymphocytes must not exceed 55% of the blood lymphocytes\r\n* CLL cells on immunophenotype (performed locally) must reveal a clonal B-cell population, which express the B cell surface markers of cluster of differentiation (CD)19 and CD20, as well as the T-cell antigen CD5; patients with bright surface immunoglobulin expression or lack of CD23 expression in > 10% of cells must lack t(11;14) translocation by interphase cytogenetics Diagnosis of recurrent CD30+ HL or CD30+ NHL, or newly diagnosed patients unable to receive or complete standard therapy OR diagnosis of relapsed/refractory CD30+ HL or CD30+ NHL with a treatment plan that will include high dose therapy and stem cell transplantation CD30 positive tumor (result can be pending at this time) Diagnosis - CD30+ HL or CD30+ NHL:\r\n* During the dose escalation phase: only adult patients with active disease failing standard therapy\r\n* After dose escalation: any patient (children or adults) newly diagnosed, unable to receive or complete standard therapy OR diagnosis of relapsed/refractory CD30+ HL or CD30+ NHL with a treatment plan that will include high dose therapy and autologous stem cell transplantation CD30 positive tumor Clinical and phenotypic verification of B cell chronic lymphocytic leukemia (CLL) and measurable disease; immunophenotyping of the leukemic cells (blood or marrow) must demonstrate a monoclonal (or light chain positive) B cell population with immunophenotype consistent with CLL (e.g., co-expressing CD19 and CD5) Known CD20-negative status at relapse or progression; CNS lymphoma or leptomeningeal infiltration Uncontrolled angina within 3 months before C1D1. Diagnosis of CLL including:\r\n* Monoclonal B-cells co-expressing >= one B-cell marker (cluster of differentiation [CD]19, CD20, or CD23) and CD5 in peripheral blood or lymph node Prior treatment with TNFRSF agonists including OX40, CD27, CD137 (4-1BB), CD357 (GITR) . Subjects must have CLL/SLL, as documented by a history at some point in time of an absolute peripheral blood B cell count > 5000/ul, with a monoclonal B cell population co-expressing cluster of differentiation (CD)19, CD5, and CD23, or if CD23 negative, then documentation of the absence of t(11;14) or cyclin D1 overexpression; alternatively patients with lymphadenopathy in the absence of circulating disease will also be eligible for this study if lymph node biopsy or bone marrow biopsy establishes the diagnosis of CLL with the above immunophenotype Immune compromised patients including but not limited to: systemic immune suppressive medications within 6 weeks of enrolling; HIV-positive and below normal CD4 lymphocytes (less than 500 cells per microliter). Patients must be tested for HIV seropositivity and CD4 lymphocyte count to be eligible for the study Diagnosis of CLL according to the National Cancer Institute (NCI)/Internal Workshop on Chronic Lymphocytic Leukemia (IWCLL) criteria or small lymphocytic lymphoma (SLL) according to the World Health Organization (WHO) criteria; this includes previous documentation of:\r\n* Biopsy-proven small lymphocytic lymphoma or\r\n* Diagnosis of CLL according to the NCI/IWCLL criteria as evidenced by all of the following:\r\n** Peripheral blood lymphocyte count of greater than 5 x 10^9/L\r\n** Immunophenotype consistent with CLL defined as:\r\n*** The predominant population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20 [typically dim expression], or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc)\r\n*** Clonality as evidenced by kappa or lambda light chain restriction (typically dim immunoglobulin expression)\r\n* Negative FISH analysis for t(11;14)(immunoglobulin heavy locus [IgH]/cyclin D1 [CCND1]) on peripheral blood or tissue biopsy (e.g. marrow aspirate) or negative immunohistochemical stains for cyclin D1 staining on involved tissue biopsy (e.g. marrow aspirate or lymph node biopsy) More than 2 x 10^6 autologous CD34+ cells/kg cryopreserved. The graft may not be CD34+ selected or otherwise manipulated to remove tumor or other cells. The graft can be collected at the transplanting institution or by a referring center SLL (lymphadenopathy and/or splenomegaly and < 5×10^9 CD19+ CD5+ clonal B lymphocytes/L [< 5000/µL] in the peripheral blood at diagnosis with measurable disease that is biopsy-proven SLL) If prior CD19-targeted therapy has been administered, subject must have CD19-positive disease confirmed by immunohistochemistry or flow cytometry since completing the prior CD19-targeted therapy. CD30 negative HL Current diagnosis of primary cutaneous CD30-positive T-cell lymphoproliferative disorders and lymphomas or mycosis fungoides Investigational or biologic therapies within 3 weeks of C1D1 Tumors must be CD20+ on prior pathologic analysis Tumor cell negative for CD20 Patients must have histologically or cytologically confirmed cluster of differentiation (CD)5 positive (+)/CD20+ B-cell chronic lymphocytic leukemia or small lymphocytic lymphoma; the diagnosis of CLL is based upon the National Comprehensive Cancer Network (NCCN) guidelines; any outside pathology slides used as inclusion criteria for the patient will be reviewed at this institution to confirm the diagnosis; the patient must meet all of the following CLL criteria to participate in this study: \r\n* Absolute lymphocyte count > 5000/uL \r\n* CD20+ and CD5+ \r\n* Bone marrow lymphocytes >= 30% \r\n* Or previous confirmed diagnosis of CLL/SLL with less than 5000/ul or less than 30% lymphocytes in bone marrow (BM) Histologically confirmed diagnosis of mantle cell Non-Hodgkin's Lymphoma with cyclin D1 overexpression by immunohistochemistry, and a characteristic immunophenotypic profile with CD5(+), CD23(-), CD20(+), and CD10(-). In tumor tissues with negative cyclin D1, evidence of cyclin D2 or D3 overexpression by immunohistochemistry will be acceptable. Patients must have a biopsy confirmed diagnosis based on a combination of histological and clinical criteria of CD30+ lymphomatoid papulosis, CD30+ primary cutaneous anaplastic large T-cell lymphoma (pc-ALCL), or CD30+ mycosis fungoides for the phase II trial; there is no specific limit or validated amount other than positive cells on immunohistochemistry (IHC) cells in tumor cells Diagnosis of chronic lymphocytic leukemia:\r\n* Lymphocytosis of >= 5,000 cells/ul; AND\r\n* Marrow aspirate with >= 30% of mononuclear cells being lymphocytes; AND\r\n* Flow cytometry demonstrating monoclonal B-cells co-expressing >= one B cell marker (cluster of differentiation [CD]19, CD20, or CD23) and CD5 Previous therapy directed against CD19, such as MAbs or MAb conjugates Patients with less than 50% donor CD3 peripheral blood chimerism on two separate, consecutive evaluations; the two evaluations must be at least 14 days apart OR patients with absolute decreases of donor CD3 peripheral blood chimerism of >= 20% if the second test shows < 50% donor CD3 cells; the two evaluations must be at least 14 days apart Patients must have persistent donor CD3 cells (>= 5% donor CD3 cells by a deoxyribonucleic acid [DNA]-based assay that compares the profile of amplified fragment length polymorphisms [ampFLP] [or fluorescent in situ hybridization (FISH) studies or variable number of tandem repeats (VNTR)]) Chemotherapy within 28 days prior to C1D1 CD123-detectable leukemia Participants must have available archival or fresh tumor tissue or both to submit to a central laboratory for CD38 assay. Expression of CD38 is measured by immunohistochemistry on fresh or archived tumor sample by central assessment using a CD38 investigational IHC assay under development: a) Stage 1: participants whose tumors are more than or equal to (>=) 50 percent (%) positive for CD38, b) Stage 2: participant has less than (<) 50% CD38+ or greater than (>) 50% CD38+ depending on the distribution of CD 38 expression of enrolled participants during Stage 2. The sponsor will advise on which eligibility criterion is permitted during the enrollment period Evidence of CD19 expression LVEF < 40% as determined by echocardiogram performed at screening or within 90 days prior to C1D1 Any level of CD56 expression will be considered sufficient for enrollment on this study Histologically confirmed diagnosis of B-lineage ALL. Verification of CD22 expression is not required Diagnosis of CD20-positive FL: CD20 immunophenotyping performed ?2 years prior to randomization Unable to generate antigen-specific MCPyV TAg-specific CD8+ T cells for infusions Primary cutaneous CD30+ disorders: ALCL and lymphomatoid papulosis Known CD20-negative status at relapse or progression Diagnosis of B-cell CLL/SLL including:\r\n* Lymphocytosis of monoclonal B-cells co-expressing >= one B-cell marker (CD19, CD20, or CD23) and CD5 in peripheral blood or lymph node AND\r\n* Bone marrow with >= 30% mononuclear cells having the CLL/SLL phenotype CD19 CAR-T cells based therapies Diagnosis of:\r\na) CLL according to the National Cancer Institute (NCI) criteria\r\nb) Small lymphocytic lymphoma (SLL) according to the World Health Organization (WHO) criteria\r\nc) MBL according to the consensus criteria\r\n* This includes previous documentation of:\r\n** Biopsy-proven small lymphocytic lymphoma or\r\n** Diagnosis of CLL or MBL as evidenced by all of the following:\r\n*** Clonal B-cell population in the peripheral blood with immunophenotyping consistent with CLL defined as:\r\n**** The population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20 [typically dim expression], or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc)\r\n**** Clonality as evidenced by k (kappa) or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. immunoglobulin heavy chain variable [IGHV] analysis)\r\nNOTE: splenomegaly, hepatomegaly, or lymphadenopathy are not required for the diagnosis of CLL \r\n*** Patients with a peripheral blood B-cell lymphocyte count of < 5 x 10^9/L and no evidence of lymphadenopathy or organomegaly will be classified as MBL; patients with a peripheral blood B-cell lymphocyte count of < 5 x 10^9/L who have evidence of lymphadenopathy will be classified as SLL; patients with a peripheral blood B-cell lymphocyte count >= 5 x 10^9/L will be considered to have CLL\r\n*** Before diagnosing MBL, CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescence in situ hybridization (FISH) analysis for t(11;14)(IgH/cyclin D 1 [CCND1]) on peripheral blood or tissue biopsy or negative immunohistochemical stains for cyclin D1 on involved tissue biopsy Confirmed positive CD70 expression on tumor tissue Washout from any prior investigational therapy of at least five times the T1/2 prior to C1D1 Washout of at least 2 weeks from the most recent radiation treatment prior to C1D1 Is within 2 months of transplant from C1D1 CD22 expression A diagnosis of CLL defined by a circulating B-lymphocyte count of greater than or equal to 5,000/uL at study entry or at any time in the past and flow cytometry confirmation of immunophenotype with CD5, CD19, CD20, CD23, CD79b, and surface Ig prior to first dose of study treatment. Have FSH levels indicative of postmenopausal state (i.e., 30-120 IU/L) documented within 21 days of C1D1. Recurrent disease:\r\n* Patient > 5 years old (yo) must have recovered CD4 count to > 350 cells/mm3 OR have disease free interval > one year from completion of cytotoxic therapy\r\n* Patients < 5yo must have recovered CD4 count to > 360 cells/mm3 OR have disease free interval > six months from completion of cytotoxic therapy Multiple recurrences are allowable as long as CD4 count or disease-free intervals have been met Histologically-confirmed CD30-positive disease; tissue from the most recent post diagnostic biopsy of relapsed/refractory disease must be available for confirmation of CD30 expression via slides or tumor block. Predominant donor chimerisms as measured by CD3 and CD33 (or other myeloid marker) Alemtuzumab or any equivalent in vivo T-cell depleting agent (or CD34+ selection) Greater than 25% of blasts must be CD33 positive. Greater than 25% of blasts must be CD33 positive. Prior to Segment B enrollment, participants on combination anti-retroviral therapy (cART) will be required to have a minimum CD4 count of >= 200 and participants not on cART will be required to have a minimum CD4 count of >= 350 to be eligible for the study; participants not currently on cART who have a CD4 count >= 200 and who agree to start cART immediately will be eligible for participation; laboratory data should be obtained within 16 weeks prior to Segment B enrollment Subjects must have CD34+ collection which allows reinfusion of ?1.5 x 106 and ?5.0 x 106 CD34+ cells/kg Possession of a CD/DVD player or ability to play a CD Recipients of CD34 selected grafts or other manipulated grafts (with any form of ex vivo T cell depletion) will be eligible to enroll if they have a CD3 count > 100; (please note: post-transplant Cytoxan for haploidentical transplants is allowable) FOR THE 31 SUBJECTS ENROLLED IN YEAR 1: Recipients of CD34 selected grafts or other manipulated grafts (with any form of ex vivo T cell depletion) will be eligible to enroll if they have a CD3 count > 100; (please note: post-transplant cytoxan for haploidentical transplants is allowable) CD4 count < 200 cells/mm^3 within 6 months of entry\r\n* Note: This refers to any CD4 obtained for routine care; documentation of CD4 is not required prior to entry CD30+ disease (result can be pending at the time of cell procurement, but must be confirmed prior to treatment with ATLCAR.CD30 cells); NOTE: CD30 + disease requires documented CD30 expression by immunohistochemistry based on the institutional hematopathology standard Required prior to infusion of ATLCAR.CD30 cells: AST =< 3 times ULN Required prior to infusion of ATLCAR.CD30 cells: Serum creatinine =< 1.5 times ULN Study 2 - CLL/SLL\r\n* Newly diagnosed (< 12 months from pre-registration on this study) CLL according to the National Cancer Institute (NCI) criteria or SLL according to the World Health Organization (WHO) criteria; this includes previous documentation of:\r\n** Biopsy-proven small lymphocytic lymphoma\r\n** Diagnosis of CLL according to NCI working group criteria as evidenced by all of the following:\r\n*** Peripheral blood lymphocyte count of > 5,000/mm^3; if present, prolymphocytes should be < 55%\r\n*** Immunophenotyping consistent with CLL defined as:\r\n**** The predominant population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20, or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc.)\r\n**** Dim surface immunoglobulin expression\r\n**** Restricted surface kappa or lambda light chain expression\r\n*** Before diagnosing CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescent in situ hybridization (FISH) analysis for t(11;14)(immunoglobulin H [IgH]/cyclin D 1 [CCND1]) on peripheral blood or tissue biopsy or negative immunohistochemical stains for cyclin D1 on involved tissue biopsy\r\n* Rai stage 0 or 1\r\n* Previously untreated\r\n* Asymptomatic with the plan for observation\r\n* Life expectancy of at least 24 months\r\n* Willing to provide tissue for correlative research purposes Patients must have a plan to receive a CD34-selected peripheral blood stem cell graft Prior treatment with CD47 or signal regulatory protein alpha (SIRP?) targeting agents (with exception of Hu5F9-G4 for patients in the Rollover cohort). Phenotypic studies (obtained within 8 weeks prior to study drug administration) from peripheral blood showing CD3+, CD57+ cells >400/mm³ or CD8+ cells >650/mm³. Pathologically confirmed MCL (in tumor tissue), with documentation of either overexpression of cyclin D1 in association with other relevant markers (eg, CD19, CD20, PAX5, CD5) or evidence of t(11;14) as assessed by cytogenetics, fluorescent in situ hybridization (FISH), or polymerase chain reaction (PCR) Radiation, chemotherapy, or immunotherapy or any other anticancer therapy (including investigational therapies) ? 2 weeks prior to C1D1. Localized radiation to a single site at least 1 week before C1D1 is permitted. Glucocorticoids within 2 weeks of C1D1 are permitted. Patients on long-term glucocorticoids during Screening do not require a washout period but must be able to tolerate the specified dexamethasone dose in this study. All patients who have received anti-CD19 directed CART therapy and completed or discontinued early from a Novartis sponsored treatment protocol that utilized CD19-directed CART cells or from any CD19 CART trial sponsored by the University of Pennsylvania with which Novartis has a contractual agreement to co-develop the CAR technology. Patients must have histologically or cytologically confirmed by the local institution CD19+ precursor B-acute lymphoblastic leukemia (pre-B cell ALL) OR CD19+ mixed phenotype acute leukemia (MPAL): a) with relapse following or refractory to at least one prior line of therapy if older than 21 years; b) in second or higher relapse or refractory to at least two prior lines of therapy if 21 years old and younger (16-21); c) or they must have a new diagnosis of pre-B cell ALL or CD19+ MPAL but are >= 60 years old and are either not a candidate for or do not wish to receive traditional induction chemotherapy Patients who were treated with blinatumomab in the past will be allowed on the study as long as they have persistent CD19 expression on leukemia cells and did not experience unacceptable toxicities with prior blinatumomab administration; patients who were treated with chimeric antigen receptor (CAR)-modified T cells targeting CD19 in the past will be allowed on the study as long as they have persistent CD19 expression on leukemia cells Clinical and phenotypic verification of B cell CLL or small lymphocytic lymphoma (SLL) and measurable disease; immunophenotyping of the leukemic cells (blood, lymph node, or bone marrow) must demonstrate a monoclonal (or light chain positive) B cell population with immunophenotype consistent with CLL (e.g.: co-expressing CD19, CD5, and CD23)